[TR] Solenoid?

[Randall]

Sounds more like a bad connection to (or perhaps inside) the battery to me.
The circuit to the starter (when you push the button on the solenoid) is
separate from the ignition circuit, so otherwise it would take two separate
faults to cause problems with both of them.

When I was a kid, that kind of stuff would happen frequently as a result of
corrosion inside the battery clamps.  So my first step would be to pull the
clamps off, and make sure that both the posts and the inside of the clamps
were shiny with a few swipes of a battery post brush.
http://www.harborfreight.com/battery-terminal-cleaner-94450.html

-- Randall