Что делать? 

Вопрос к знатокам английского во всех его нюансах,, включая фонетические.

и что такое high-fall?

High Fall же - это такая интонация высокого падения в конце утвердительного предложения. Мы ее  на 1-м курсе иняза учили.  Она абсолютно нормальна в русском языке, но, возможно, звучит слишком категорично в английском.

https://usefulenglish.ru/phonetics/intonation-and-tones

помнится, была у нас на факультете такая лингафонная лаборатория, там слушали и повторяли, слушали и повторяли... а уж с теперешней техникой никаких лабораторий не надо - скачать, слушать и повторять :-)

при этом хороший музыкальный слух очень помогает

Пока могу лишь посоветовать вам почитать книги типа "Get Rid of Your Accent".

В СССР вопрос начали детально разрабатывать покойный Александ Давидович Швейцер, Джульетта Аршавировна Шахбагова и ее коллега и подруга (запамятовал фамилию). Их книги - на сети, а диссертации - в Ленинке.

Как избавиться - слушать американскую речь, смотреть фильмы, подмечая особенности и пытаясь их копировать.

The best way to learn to speak a foreign language like a native isn't from a book, but by living/working amongst native speakers. However if, like most of us, you can't afford to live abroad just to perfect your accent, the next best way is probably to listen to recordings of foreign TV programmes on YouTube -- for example, news broadcasts, as news presenters tend to speak clearly and have 'neutral' (non-regional) accents.

For American accents try searching for the NBC/Fox News/RT (America) etc channels on YouTube and specify "subtitles" in your search. Choose programmes with subtitles written by humans rather than by computer, as computer-generated subtitles can contain lots of errors. For British accents search for BBC News/Sky News/Euronews/RT (UK) etc. Aljazeera also employs mainly US and UK newsreaders.

Some politicians and businessmen -- but not Donald Trump or Boris Johnson! -- are also very clear speakers with fairly 'neutral' accents. Here, for example, are Barack Obama and Bill Gates (with subtitles) on YouTube's 'Learn English Online' channel:  https://www.youtube.com/watch?v=Nn_iCKa7neM

Note that you don't have to agree with anything that these news broadcasts or politicians say -- you're listening to them simply to learn the English accent. Preferably choose programmes with presenters who don't speak too quickly.

Firstly see:

https://www.youtube.com/watch?v=_HGxfR7Sziw

(Apologies for the 'tinny' audio).

Then, for a demonstration of how these 7 different forms of intonation are used in English to change the meaning of a word/phrase/sentence -- for example, to change it from a statement to a question, or to indicate surprise, sarcasm, doubt etc -- go to

https://www.youtube.com

and search for:

"English" "intonation"

(as written, with quotation marks).

It will then present you with a list of resources (short lessons and exercises) on US and UK English intonation, explaining how and when to use high-fall, high-rise etc. (Ignore the videos by 'ETJ English', which imho don't explain the subject very well).

Normally, I consider attempts to "speak like a native" not worth the trouble. But this was an interesting case, and I will see where johnstephenson's resources will lead me in my inquiry.

Thanks and have a good day

Ulkina, читайте первые полторы строчки Джона 22:20 там все правильно. Остальные - NBC, CNN, Barack etc. - лирика. Это только в советских фильмах разведчики разговаривают на немецком с баварским акцентом (((

Igelschnäuzchen никогда не слышала, звучит как высморкаться (sich schnäuzen)

https://www.quelle.de/mode-freizeit/alpenwelt/rund-um-die-tracht/bayerisch-fuer-anfaenger/#els=true&sview=extern

там не только интонация, там и слова другие, не говоря уж о том, что там черт ногу сломит :-)

"Всего в Баварии более 60 различных диалектов... Помимо собственно баварских диалектов, в Баварии распространены также франкские (восточно- и южнофранкские) и алеманские (швабский) диалекты южнонемецкой группы диалектов"

После ссылкам johnstephenson прошла, но  вопросов  осталось больше, чем ответов. 

1. Вот интересное видео из предложенных   

https://www.youtube.com/watch?v=gzHmWg3ylFg

Правда, здесь обратная ситуация - uptalk. 

2. В другом видео я увидела, что падающий тон означает команды. Может, это напрягает американцев, что с ними разговаривают тоном, смахивающим на команды?

3.  Да и вообще, присмотрелась я к этому высказыванию (см.ниже), и действительно запуталась. С каких пор flat tone means angriness?

"Exaggerate your changes in intonation a little. (Sorry, but Russian accents usually sound angry to Americans."

1) I don't know why your американец says that "Russian accents usually sound angry to Americans", and I think it's an (inaccurate) generalisation. I think it's more accurate to say that "some Russian speakers' intonation can sometimes sound like a statement/command, rather than a question".

That's because, overall, the intonation in spoken Russian tends to be slightly flatter than in spoken English. It doesn't mean that falling intonation in English always indicates anger or a command; in fact, falling intonation is extremely common in spoken English when you're simply making a statement (as well as when giving some commands). Therefore I would ignore this comment by your американец and definitely wouldn't worry about it. Instead, try to watch and listen to native US/UK English-speakers, and learn from/copy them instead.

2)  The AccurateEnglish clip which you've included a link to at 15:31, is slightly different as it's purely about so-called 'uptalk' in English (also known more technically as 'rising [terminal] inflection'). 'Uptalk' began to enter UK English about 20 years ago, from America. It seems to have originated amongst a few Hollywood actresses a few decades ago and has since been copied by a few teenagers -- mainly girls -- in both the US and the UK, who have seen and heard these actresses on film/in interviews.

However, many people in both the US and UK (including me) find 'uptalk' very irritating, as the clip explains very well. This is because it appears to turn a simple statement into a question, leaving the 'other person' confused as to whether the speaker is making a statement, or asking a question. That's why the presenter in your clip at 15:31 says that inappropriate 'uptalk' tends to make the speaker sound immature and doubtful about what they are saying. Therefore I would avoid using 'uptalk' altogether unless you want to ask a question, as 99% of Americans and Britons don't use 'uptalk' for simple statements.

3) Note that these are just general guidelines. There are some parts of the UK where the intonation is slightly different from the rest of the country -- eg, in the Liverpool area (in the "Scouse" accent) and the Birmingham area (in the "Brummie" accent). No doubt the same applies to the US (where there are even more regional accents) and to Russia. However, I suggest that you avoid learning to speak in one of these strong regional US/UK accents, and concentrate instead on developing a more 'neutral' accent and intonation, such as most of those in the YouTube clips I suggested.

It is quite the opposite of what I (and Sjoe!) originally thought.  The amazing thing is that none of these subtleties ever occured to me:  the   "uptalk" video does say that checking on the intonation is our last effort in mastering English. 

Johnstephenson, can I subscribe to your channel?

I don't have a channel -- but if this is the only aspect of English that you haven't yet mastered, you've probably nothing to worry about and are virtually a native English speaker.

For an explanation of the background (if interested), go to the Russian Wikipedia (https://ru.wikipedia.org/wiki) and search for 'Башни Фолти (телесериал)'.

For the clip itself, see: https://www.youtube.com/watch?v=tcliR8kAbzc

Russians don't normally talk like that, do they....?