I would be looking for a bad, corroded, or missing ground between the battery and the body.
Did you remove the battery cables and cleaned the posts and terminals, or did you just chemically clean the posts? Chemically cleaning doesn't always get the job done, especially if they are really bad. I would also consider replacing the terminals on the cables or replacing the entire cable. A lot of problems on old stuff is fixed with good clean metal to metal battery connections, on both ends of the cables. Gene
Thanks for the suggestions. I will try and check the grounds tomorrow. I checked for continuity, resistance, and voltage drop between the battery and the fuse/relay box under the hood and everything checked out fine. I physically cleaned the battery posts and terminals. I don't think the issue is at the battery, but a ground problem at the relay boxes sounds like a possibility.