The AM503 I have does drift pretty badly, I'm not sure how much the probe does though, as I've normally let them warm up together. This is one area where the more modern stuff is better, that's true. Still usable though if you let it warm up for a substantial period of time and check occasionally when you're using it.
Might experiment one day to see if I can tell how much a probe alone drifts. Making my own amplifier is on the to-do-one-day list - should be reasonably practicable unlike repairing or building a probe.