Thanks so much for your helpful responses! And thanks for the kind words BSUPC.
I started with the easiest fix and figure I'll work up from there. So far I bought the special brush and cleaned the terminal posts and clamps. I added the felt washers and bought some grease/lube that should help the connection stay solid for a while.
Happy to report it's started fine ever since! Hope this remains the case. If not I'll get into the more detailed fixes you guys gave.
Thanks again!