There are a lot of very low power FM, 88 - 108 MHz band xmitters out there (as you have seen) for very little money.
One I have used and was quite pleased with (leatherback turtle hatching monitors):
https://www.canakit.com/mini-wireless-fm-transmitter-kit-ck105-uk105.html
Their output level is legal (per Part 15 FCC regs) so long as you do not alter the antenna.
The provided antenna could be loosely wound within the rocket body (the xmitter will easily fit as well) and would give you, I believe, an adequate propagation pattern for the range and orientation you describe. Or the antenna could be simply trailed behind the rocket, giving a better range. Its drag, however, might affect total payload weight allowance.
The drifting problen can be settled by replacing the LC circuit, that sets the freq, with a crystal. You would, of course, want to choose a freq to so the output is within a "dead-zone" for your area, for neighborliness, if nothing else.
They usually come with a mic modulator attached. If you use the TTL sig to drive a small oscillator (1kHz, or whatever), its output could be used to drive the modulation of the xmitter.
Now that I think about it, you might be able to get by with a simple 10 second digital recorder/playback device (like in a birthday card where you record your own message) set up with a timer to continuously loop. Feed the output directly into the xmitter.
These little cheapos can run for days (up to a week) off of one AAA.
There might be a range issue, but I would think that if your visually tracking the rocket, you could head in the direction of landfall and, even if heavy foliage, eventually pick up the signal up to locate the rocket.
All the while staying Part 15 compliant.
And I would suggest putting the real money into a very good receiver/antenna rig.