Let me start my Answer by explaining that It's not possible Moshe could have been born on 'אדר א and be niftar on 'אדר ב, for two simple reasons #1. Moshe Rabbinu was niftar at 120 years old. #2 He lived a complete (full) number of years. (Meaning he passed away after a full 120 years to the day of his birth (on ז' אדר). If Moshe passed on 'אדר ב that would be an un- full 121 years.
Perhaps what you always thought, (And some Midrashim do in fact say- (it's a Machlos) was that Moshe was born on a Regular ז אדר and he was niftar on a "Leap Year" with an אדר ב, and he passed on 'ז' אדר ב. That's considered "A Full 120 years" (As the שולחן עורך brings down the Halacha by a Bar Mitzva. A kid born Reg. Adar who turns 13 on a Leap Year becomes a Bar Mitzvah on the 2nd Adar. Interestingly, this case works the opposite way as well.)
Now, the bottom line is that it's an Argument when Moshe died.The Pasuk. in V'Zot HaBerachah at Moshe's passing says: וַיִּבְכּוּ֩ בְנֵ֨י יִשְׂרָאֵ֧ל אֶת־מֹשֶׁ֛ה בְּעַֽרְבֹ֥ת מוֹאָ֖ב שְׁלֹשִׁ֣ים י֑וֹם וַֽיִּתְּמ֔וּ יְמֵ֥י בְכִ֖י אֵ֥בֶל מֹשֶֽׁה׃
Some Midrasim explain: After his death that Beni Yisrael mourned for Moshe Rabbinu for 30 days, and then had Hespedim for him for another 30 days, totalling 60 days in which case Moshe passed on a Leap Year on אדר ב. Other Midrashem hold they Mourned for 30 days ( Hespid included in that time), and he passed on a Regular אדר.
Now, According to this Midrash VeHizhir that "Oliver" brought and "Double AA" found out was brought down by Rabbi M. Kasher who wrote a "Likud" (gathering) of various Midrashim, in his Sefer titled Torah Sheleima, that your Rav told you about, were it says that Moshe picked Yehoshua to help him lift his hands so Beni Yisrael will be victorious over Amalek, because they were both born on אדר ב, then the Midrash VeHizhir can hold either way. Either that Moshe died on a Regular ז אדר or on a "עיבור שנה" (Leap Year) on 'ז' אדר ב.
Hope I helped. Good Shabbos.
P.S. If you- Or anyone elese on this Page want to understand the Machlos in detail, with a reconciliation I added this Link to the Chasam Sofer.(https://www.sefaria.org/Deuteronomy.34.8?with=Chatam%20Sofer)