In the old days, chirp was a favorite discussion topic.

You have been given a number of ideas, so here are a few more. When the
feedback capacitor dielectric heats up, the frequency can change. Using an
air dielectric is best, but a silver mica or NP0 capacitors usually work.
If you do not want to regulate the screen voltage on the oscillator, you can
adjust it until you find the "sweet" spot, which will reduce drift. You
need a tapped resistor or rheostat to do that.

Crystal oscillators are supposed to be simple, so the cure should be simple,
too. Something is heating up causing a frequency change when you key it.

Make sure you are not overloading your receiver, causing a phantom chirp
(only one can hear it is you).

73, Colin K7FM