The seventh month is where the Haab's relation to the sun is most often misread, because the name means new sun and invites the assumption that it marked a solstice. It did not, and could not have for long: without a leap-day correction the 365-day count slips against the tropical year by roughly a day every four years, so any seasonal fit the name records was true of a particular century and drifted away from it afterwards.
The Maya were entirely capable of tracking the true year. The Dresden Codex contains tables of Venus and of eclipses with corrections applied over long intervals, and Classic architecture at several sites is aligned to solar events. The Haab was simply not the instrument used for that work, any more than a week is used to date an equinox.
A Maya date does not name a month and a number in the way a European one does. It names a day in the 260-day Tzolk'in, by number and sign, and a position in the 365-day Haab, by day and month, and the two together form the Calendar Round, which repeats only after 18,980 days, a little under fifty-two years. Within the Haab the days of a month are numbered from zero to nineteen, and the zero day is not called first but seated, the seating of the month, a usage the Classic inscriptions write with its own glyph.