Hi - there are some modes when the 737-300 will use trim by itself - not very much unlike the infamous MCAS!
There is a "mach trim" system at very high speeds, that adds some trimming to offset the center-of-lift movement at higher Mach numbers
You are probably seeing the "speed trim" system in action that works when flaps are not up AND N1 > 70% (iirc), so you will see it mostly after take-off.
It automatically adds trimming to maintain the current airspeed - so if your nose rises and the speed falls, the plane will trim "nose down" to avoid speed going too low. Most pilots get confused about this, because when they push down on the yoke to accelerate after take-off, the plane will trim "nose up" to get back to the initial speed! Pushing the yoke against this will cut out the trim, but letting go will enable it again!
Trimming with the trim switches will "reset" the target speed to the current airspeed. Obviously raising the flaps will also help - or reducing thrust (for level flight, for example).
Let me know if this is what you are observing?
Cheers, Jan