It does sound like the issue is the motor isn't slipping on the drive belt as it needs to when draining.
With these machines, the design is that, when draining, the motor drives the pump strongly and the pulley for spinning the tub <span style="text-decoration: underline;">weakly.</span> If the belt from motor to transmission pulley can't slip under load, the motor will try to spin the whole drum full of water and that overloads the motor. The design is that once the drum is empty of nearly all its water and the tub can spin easily, the slippage will reduce and the spin will get up to speed.
I'm not familiar with this particular machine, but in Australia we had a Simpson machine that used a similar system, but a bit more basic mechanism than the Maytag. If the spring tension is adjusted too strong (not sure if Maytags have a similar tensioner spring) then the belt can't slip when it needs to, the motor overheats quickly and goes up in smoke.
As I understand it, the Maytag motor moves on a sliding "carriage" referred to above, if the carriage is sticky, jammed or not adjusted right, the motor won't move back enough to let the belt slip when needed. You need to check this and get it adjusted right before trying to spin it again. Do NOT keep trying to make it spin without finding and fixing the fault - you will burn out the motor. The motor has two windings, start and run. The start winding only operated for a second or two to accelerate the motor up to speed quickly. If something is preventing the motor reaching full speed in one or two seconds, the start winding will burn out - the start winding pulls a lot of amps and can only run for a few seconds, once the motor gets up to full speed the centrifugal switch disconnects the start winding and it cools down again, with the motor continuing on just the run winding.
The information you have given leads me to think this is the issue - the smell and dimming lights in the house indicates that the motor is overloading and overheating, and the fact that it gets up to speed OK once you have baled out the excess water suggests the motor works OK once it isn't being overloaded. (So I don't think the motor is burned out...yet...)