So from what I understand of your posts is that the battery contacts get hot between battery and motor while the motor turns slow but does turn?
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After I installed the new battery and charged it fully, the motor ran smooth-ish, but slowly, at the first test run lasting just a few seconds. I switched it off and back on, and it just made a buzzing noise and did not run at all. I tried a couple more times, each lasting just a few seconds, and I got the same buzzing/no spin result and noticed the area where the motor is located got very hot. I noticed the display was now telling me to recharge the battery (even though it had just finished recharging just a few minutes ago). I then took off the housing, and indeed the motor was hot. With no housing, I switched on the toothbrush and saw it struggling to spin in fits and starts. When I helped it along using a finger to spin the pinion, it would spin one or two turns only. I decided to test the battery strength by directly connecting the battery to the motor terminals by fashioning a couple of pieces of jumper wire that are a little thicker gauge than that of a small paper clip, bypassing the circuit board entirely. Within a few seconds of making the connections, the wire I was holding burnt my fingertips. Using a multimeter, I tested the battery to have 1.29 v. (Today, with the battery fully charged, I tested it to have 1.45v. It is an "A" cell Ni-MH with 2700mAh.)
As you can see from the video, removing the motor from the frame would be quite a challenge, which I have yet to figure out.