Trailer Feels Heavier After Hauling for Period of Time

This is easy to check by disconnecting the trailer wiring when it starts to act up....on a side road with no traffic. Also by letting off the gas if it does not coast like it normally does, then it's likely the brakes.

Bad trans will slip, not lug the motor.
Bad engine will be bad from start to stop, not act up in the middle of a tow....usually. This is ofcourse assuming all needles are where they should be- Coolant, oil, trans temps, etc.