I've heard that the bank may agree to a "one time adjustment" to lower the payments on Mortgage #2 because of paying a very large payment. Is this something that really happens?
It's to the banks advantage to reduce the payments in that situation. If they were willing to loan you money previously, they should still be willing. If they keep the payments the same, then you'll pay off the loan faster. Just playing with a spreadsheet, paying off a third of the mortgage amount would eliminate the back half of the payments or reduces payments by around two fifths (leaving off any escrow or insurance).
If you can afford the payments, I'd lean towards leaving them at the current level and paying off the loan early. But you know your circumstances better than we do.
If you are underfunded elsewhere, shore things up. Fully fund your 401k and IRA. Fill out your emergency fund. Buy that new appliance that you don't quite need yet but will soon. If you are paying PMI, you should reduce the principal down to the point where you no longer have to do so. That's usually more than 20% equity (or less than an 80% loan).
There is an argument for investing the remainder in securities (stocks and bonds). If you itemize, you can deduct the interest on your mortgage. And then you can deduct other things, like local and state taxes. If you're getting a higher return from securities than you'd pay on the mortgage, it can be a good investment. Five or ten years from now, when your interest drops closer to the itemization threshold, you can cash out and pay off more of the mortgage than you could now.
The problem is that this might not be the best time for that. The Buffett Indicator is currently higher than it was before the 2007-9 market crash. That suggests that stocks aren't the best place for a medium term investment right now.
I'd pay down the mortgage. You know the return on that. No matter what happens with the market, it will save you on interest. I'd keep the payments where they are now unless they are straining your budget unduly. Pay off your thirty year mortgage in fifteen years.