Moshe was 80 when he first approached Par'o and the makos lasted a year, so he turned 81 in the Adar before the Jews left Mitzrayim. As he turned 120 in the Adar before the Jews entered Israel, the Jews were in the desert only 39 years. What gives?
Seder Hadorot tells us that Moshe Rabbeinu was born in the year 2368.
It then tells us that G-d spoke to Moshe at the burning bush on the 15th of Nissan in the year 2447. This would mean that Moshe had just turned 79 a little over a month before, on the 7th of Adar. Moshe then goes to talk to Pharoah when he is 79.
(Incidentally, the Seder HaDorot brings many different opinions of when each of the plagues happened, but see the GR"A on Seder Olam, who says that the 12 months the Mishna says Egypt was plagued starts from G-d appearing to Moshe at the burning bush.)
The Exodus from Egypt was a year later, on the 15th of Nissan in the year 2448, when Moshe had recently turned 80.
Moshe then passes away 40 years later on the 7th of Adar, in the year 2488. Moshe was 120 years old and this was the 40th year the Jews were in the Desert.
The question remains: Why would the Passuk say the Moshe was 80, if he was only 79? I'd have to answer, Miktzas Shana K'shana, part of a year can be considered as a whole year (Rashi in Devarim 13:33 uses this logic to calculate how people could have died in the desert before they were 60). Since Moshe had already started his 80th year, the Torah calls him 80.
One possible problem with this. Seder HaDorot tells us that Aharon was 3 years older than Moshe, and born in 2365, based on the passuk quoted in the question. If Moshe was 79 when he visited Pharoah, would we have to say that Aharon was really 82, in his 83rd year? I guess this would depend on which day Aharon was born, which I did not see written in the Seder HaDorot.
The Kehot Interpolated Chumash says explicitly (bold is chumash text, regular is interpolated commentary):
However, the source the are quoting from (Likutei Sichot vol 20, p. 26, note 15) doesn't actually say 10 months, it says "less than 79 and a half". The Lubavitcher Rebbe is using our verse (Shemot 7:7) to prove that someone can be called a certain age, once they have entered that year. In our case, Moshe was called 80 even though he had only entered his 80th year, but was still 79.
It appears that the Kehot Chumash says 10 months because the Lubavitcher Rebbe (in Likutei Sichot vol 6, p. 59, note 18) says that according to the simple understanding of Rashi (Shemot 7:25), as well as the Midrashim that Rashi was based on, there were 3 weeks of warnings before every plague (except Makat Bechorot), even the ones where no warning was mentioned. The Kehot Chumash on verse 7:25 says:
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Moshe was 79 plus when he came first to pharaoh. His 80th birthday preceded the Exodus. He died about 40 days before Israel enterred the land which was exactly 40 years after the Exodus. When a person is past 79 he is in his 80th year, which is what the Torah refers to.
It is possible that when the Chumash says Moshe was 80 that it either is rounding (79 rounded up to 80) or that it does not mean right then, but that at some point in the year long process of speaking to Pharaoh, Moses was 80 (that potion being the final 5 weeks in Egypt, when admittedly, Moses really wasn't speaking to Pharaoh).
The Da'as Zkeinim (and the Chizkuni) at the beginning of Parshas Noach addresses this issue in a different context - The posuk says תמים by Noach, and the Midrash says (Bereishis Rabba 30:8) that anyone described as such lived to an age the which is the multiple of 7 (full שבוע). Noach's 950 do not add (or divide) up. He answers that he lived this amount from the time which the Torah described him as תמים, the 350 years after the Mabul. However, there is an extra year, the year of the Mabul itself, which came after he was described as תמים! Says the Da'as Zkeinim:
So a year in which the laws of nature are suspended does not count.
This answers a very similar question to your own:
the Torah says that Noach was 600 at the start of the flood (Bereishis 7:6), and that he lived 350 years after the flood (Bereishis 9:28), and that he lived a total of 950 years (Bereishis 9:29). So what happened to the year of the flood? It isn't counted, as above.
Similarly by the age of Moshe, the year of the Makkos was not counted as a year of his life, as it was a year in which the laws of nature were not firmly in place (I can't find the medrash which says this explicitly, but I have a team of scholars working on it). Thus, Moshe was still 80 for the count of his years when they left Egypt, and there were 40 years in the desert before he died at 120.