When you cleaned your battery terminals, did you also do the OTHER end of each of the cables?? That would be a good thing to do. Also the big ground wire from the transmission to chassis. Also look for corrosion where the wire end lugs are crimped onto the wire itself.

Before you remove the other end of the big (+) cable, disconnect the (-) cable at the battery.

How old is your battery anyway? Maybe it's time for a new one.