The power light, which is known as the "operation" light on this particular model, stays on when I turn it off and there is a light hum that sounds like a motor running coming from the tape deck area and tapes sound slightly shaky when I play them and records one speaker. What could this be?

I opened it and checked everything and still cant figure out the problem.

maybe it's the hum of changing opinion...

It could be a broken leaf switch. 

Most likely, somewhere in the deck.

Or... someone, who tried to fix it before,  shorted that broken switch in the power circuit.

There are many types of these switches, but those from the decks, more often look like this one

just in case

http://www.ebay.com/bhp/leaf-switch

i agree with metad,but try the usual switch cleaner first in all the switches