I have been made aware of the proper (original?) pronunciation of some of the Hebrew letters from an old Sephardic Jew. For example the ashkenazic "suf" should be pronounced with a "th" as in the word "thick". Would there be problem with changing the pronunciations I use during davening? Does it matter if I switch back and forth - at least until I get used to it?
There is no such a thing as proper pronunciation. Ashkenazim view their pronunciation as proper, Sephardim think theirs are proper, Yemenites think their own is the best.
It turns out that all modern (non-Samaritan) pronunciation traditions stem from three main historical pronunciations.
The Gaonic Babylonian pronunciation is preserved quite well by the Yemenite Jews. This system has 6 distinct vowels.
"Sephardi" pronunciation is the term for the group of pronunciations used by Jews outside of Europe and Yemen. This is far from a homogeneous system but they all have 5 vowels. This system came from the so-called "Palestinian" system.
The Tiberian system, which was the other main medieval pronunciation system had 7 distinct vowels. It is this system that was authoritative during the medieval period. As such, Tiberian niqqud was preferred over Babylonian and Palestinian systems and is used until this day.
The Ashkenazim used to speak in a variant of the Palestinian system until almost a thousand years ago. At that point, something happened and they transitioned to a Tiberian-influenced system.
The antiquity of the Tiberian system has been established by modern scholarship. Still, it has some features which are relatively late (e.g. short a in a closed syllable -> short i; this is seen in words like Miriam which was originally Mariam).
But to seek an original pronunciation is fruitless. It is clear that many systems of pronunciation existed even in the First Temple period (cf. Shibboleth incident). Hebrew pronunciation is also always changing.
If you consider Tiberian to be authoritative, then Tiberian is the way to go. Few people know it or learn it nowadays because it is a very difficult system to learn.
Summary of Yalkut Yosef 685.20 Footnote 17:
Harav Yaakov David ben Ze'ev in Bet Ridbaz #27, Harav Baruch Epstein in Mekor Barukh v2, pg397b, and in responsa Divre Yosef Schwartz p167a wrote that the sefardic pronunciation of hebrew is more authenticate than the ashkenazic variant. There has been some recent pushback with regards to the pronunciation of certain vowels as being more authentically pronounced by the Ashkenazic Jews. But even this has been refuted, see Yabia Omer V10 OH#10.
Harav Natan Adler, hired someone to teach him the proper sefardic pronunciation, after which he switched and maintained the sefardic pronunciation for all his prayers. This has been testified by sefer Tzeror Hahayim and Derekh Hanesher p45a. But Harav Natan Adler did not suggest others emulate him.
Harav Mazuz added the following: Ashkezanic Jews should be concerned with their mispronouncing the letters "het" and "ayin" improperly. One can be saying "you shall destroy Gd" instead of "you shall serve Gd" or "Gd was taken captive" instead of "Gd swore" and see many such examples in sefer Sefat Emet pg162 by Harav ben Tziyon Cohen.
From this you can learn the following:
Would there be problem with changing the pronunciations I use during davening? It would seem from the above, that no its not a problem since if Rav Natan Adler did it, you can surely do the same as long as your intention is correct. The Magen Avraham objected to make these changes because of Minhag Avotenu. In this case, however it might not apply.
Does it matter if I switch back and forth - at least until I get used to it?
Yes it could be a problem. Of course this does not mean that while you are learning and making mistakes. Since that is part of the process of learning. The Torah was not given to angels nor does Hashem expect perfection only to put all your effort into it.
Switching back and forth is however a problem, if your intent is one time to say the prayer in ashkenaz and then another in sefardi. Because whichever pronunciation you are using, as long as your custom follows this you are at protected, since you are following the tradition of your fathers. But when switching back like that what are you relying on? You are deliberately putting yourself in a position of mispronouncing as above by Rav Mazuz. If now you accepted that the sefardi is the correct pronunciation, why are you switching back? If you reject the above Rabbeim saying that the sefardi pronunciation is not the correct one, then why are you forsaking the tradition of you fathers?
See the following series by Rabbi Daniel Glatstein for further information on this subject "Ashkenazic Vs. Sefardic Pronunciation - Which One is More Correct?":