While looking through the posting, it seems that there needs to be a summary update to make things clearer. I will explain in my own words and then the original with citations may be clearer.
The OP says that it appears that the ark floated for 150 days, but that was five months and asked how this can be as it implied 5 straight 30 day months. That is, the mabul started on the 17th of the second month and the ark touched down on the mountain on the 17th of the seventh month, while Noach 8:3 seems to say it was 150 days.
The short answer is that the meforshim explain that the 150 day reference does not mean that the ark touched down after 150 days and that the 150 days were 5 months. The detailed explanation follows.
The first note is that, while we assume that the flood began in Cheshvan as the second month, so that we count the first month as Tishrei, Rashi on Noach 8:5 cites Rabbi Joshua that the numbered months start from Nisan and the flood started on the 17th of Iyar. However, this does not change the analysis.
The Mabul began on 17 Chevan 1656. The torah says that the period of the Mabul ended on the 27th of Cheshvan after a complete Solar year (365 days). This means that the lunar year (based on our current calendar) was 355 days, with 30 days in Cheshvan and 29 days in Kislev.
The torah references three periods of time. 40 days, 150 days, and 150 days for a total of 340 days. The remaining 25 days are the final days when Noach was waiting to be instructed to leave the ark.
The first forty days was the period of the actual flooding and the rain. After 40 days, the rain stopped.
The next period of 150 days Noach 7:24 is called the period when
כד וַיִּגְבְּרוּ הַמַּיִם עַל הָאָרֶץ חֲמִשִּׁים וּמְאַת יוֹם:
24 And the water prevailed upon the earth a hundred and fifty days.
Rashi on Noach 8:3 calculates the exact time this occurred.
This period started on 27 Kislev, 40 days after 17 Cheshvan. Rashi gives the exact time as
27, 28, 29 Kislev - 3 days
Teves - 29 days
Shvat - 30 days
Adar - 29 days
Nisan - 30 days
Iyar - 29 days
Total - 150 days
The next period is called the period of abatement (or diminishing) and begins on 1 Sivan. Note that this is not necessarily a period of 150 days. That depends on how 8:3 is translated. Rav Hirsch translates Noach 8:3 as
and after the end of 150 days, the waters were abated
If this is true, then the second period of 150 will be as follows:
Sivan - 30 days
Tammuz - 29 days
Av - 30 days
Elul - 29 days
Tishrei = 30 days
Cheshvan - 2 days
TOTAL - 150 days
Noach exited the ark at the command of Hashem 25 days later on 27 Cheshvan.
Thus, this period starting on 1 Sivan ended on 2 Cheshvan. On 1 Tishrei as specified in 8:13, Noach saw that the ground had dried up. However, it took until 2 Cheshvan for the water to completely drain off and until 27 Cheshvan for Hashem to tell Noach to leave the Ark (8:14)
Rashi seems to say that the second mention of 150 days is the same as that specified in 7:24. The end of that period as mentioned in 8:3 is the beginning of the period of diminishing as translated by the Chabad website. Thus, the period of diminishing lasted from 1 Sivan to 1 Tishrei when Noach saw that the ground was dry. He then waited until 27 Cheshvan for the ground to be completely dry and Hashem to tell him to leave the ark.
Now dealing with the specific dates of the Ark resting on the mountain on the 17th day of the seventh month, and the tops of the mountains appearing on the first day of the tenth month.
Rashi on Noach 8:4, says that this means that the seventh month was counted from the end of the rain. Thus, the seventh month from the 27th of Kislev is sivan. That is the first period of 150 days ended on 29 Iyar and the next period (of diminishing) began on 1 Sivan. The ark then came to rest on 17 Sivan.
Rashi then says (Noach 8:4-5) that the reference to the mountain tops appearing on the first of the tenth month actually means the tenth month counting from the start of the flood on the seventeenth of Cheshvan. This means (according to Rashi) that the tops of the mountains appeared on 1 Av. Rashi on 8:5 explains why this is so, based on the description of what happened with the raven and the dove. If the time was calculated from the end of the rain (27 Kislev) then there would not have been sufficien time for the events at the end of the flood to have happened and for Noach to have seen that the ground was dry on 1 Tishrei.
According to those who would say that these two references are also calendar references, then the Ark came to rest on the mountain in Nisan (the seventh month), but this was still in the middle of the 150 day period of the water being above the earth. This period ended on 1 Sivan (the ninth month), but the mountain tops appeared on 1 Tamuz (the tenth month).
Rav Hirsch seems to say that the time for the complete abatement was 150 days, but the ark touched the top of the mountains on the seventeenth day of the seventh month (Which is Nisan counting from Tishrei). The complete abatement was on Rosh Chodesh of the tenth month (Rosh Chodesh Tamuz).
In order to say when the 150 days started and ended, one would have to calculate from when the flood started, 40 days and nights of flood, 150 days of "waters swelled (7:24) and 150 days of abatement. During that 150 days, the ark touched down and then the water continued to abate. In any case, it is not a matter of 150 days = 5 months necessarily.
Another explanation, could be that since the new moon was not visible all this time, there were 5 months of 30 days. Once the moon became visible again, there would be 29 day months (the minimum allowed) until the calendar became in sync agin with the actual moon.
Rashi would say you are making a mistake as to what the seventh month in 8:4 means. I will check other meforshim and update this after Shabbos.
in the seventh month: Sivan, which is the seventh counting from
Kislev, in which the rains stopped. — [from aforementioned source]
That is, the rains stopped on the 27th of Kislev, and after 150 days started diminishing which was 5 months and 3 days (the first of Sivan). In the seventh month after Kislev, (Sivan) on the 17th day of that month, after 17 days of diminishing, the ark came to rest.
on the seventeenth day: From here you learn that the ark was submerged
in the water eleven cubits, for it is written: (verse 5) “ In the
tenth [month], on the first of the month, the mountain peaks
appeared.” That is [the month of] Av, which is the tenth month
counting from Marcheshvan, when the rains fell, and they were fifteen
cubits higher than the mountains. They diminished from the first of
Sivan until the first of Av fifteen cubits in sixty days, at the rate
of a cubit in four days. The result is that on the sixteenth of Sivan
they had diminished only four cubits, and the ark came to rest on the
next day. You learn [from here] that it was submerged eleven cubits in
the waters [which were] above the mountain peaks. — [from