intermittent vibration, humming, shaking

What was the light reset for? Was the issue that caused it repaired or just the light reset?
The description could be almost anything from a bad CV joint to engine misfires to transmission or bad brakes.
When it starts do you feel it more in the steering wheel or the seat? Does it change if you let off the gas or lightly ride the brake? How about when you steer, does it get louder turning one way over the other? If you turn fully left and fully right at low speed in a parking lot is there any noise like clicking or popping?

Because it could be a number of things, the biggest problem seems to be with acceleration, driving roughly/vibrating/humming for about ten minutes, and then okay until the next acceleration. Nothing noted with turning, clicking, popping. I feel it in the gas pedal and steering wheel. And possibly some faint smell of burning. The reset might have been needed because I topped off with the car running, although in the last few months if I have reset it three times, cannot remember if the car was running when I topped off. 145,000 and no known tune up, wondering about the fuel injectors or fuel filter/spark plugs/wires/coil? Less of a problem at slower speeds or idling.

It sounds like you need to have it scanned to see what the codes were and if there are any hidden codes. Then possibly do a full tune up. Resetting the light does not repair the problem. If you can post the code(s) we might have a starting point.

Have you had any changes or updates with your vehicle problems?