Here's a bit that might help with the connections. I usually start diagnosing battery / cable connections at the battery. Sometimes crud is growing on one of the cable ends right at or near the battery. Once your battery connections are clean, di-electric grease on the cable ends and battery terminals will help seal up the connections and hopefully keep the connections clean. You probably know this.
Check the ground that goes from the battery to the engine block, at the engine block. Is it loose or is there anything blocking current flow? Both the block and cable end should not have paint, rust, or other corrosion. I usually use some di-electric grease here as well.
If you still can't find the point(s) of high resistance you might think about doing voltage drop testing. You likely know how to do this. If not, look it up online. You would probe across two connections while someone else tries to crank the engine. I would not crank all that long since you probably have something getting very hot.
Sometimes vaseline will serve the same purpose as di-electric grease. Engine heat can cause the vaseline to heat liquify and find its way out. Di-electric grease does not seem to have this problem.
As for the battery drain, do you have an amperage clamp for measuring amperage draw?
Paul Lohr