Hi Bruce,
I would actually start with the battery and connections. What you describe sounds like a weak battery or corroded connection or cable sapping power to the starter.
The '67 had twin 6 volt batteries hooked in series to create 12 volts . The number of cables and connections with those as well as the limited availability of the right 6 volt replacements created issues. Add to this that most LBCs tend to "leak" power even standing idle, the batteries drain and damage unless disconnected.
Most of us added disconnect switches to protect the batteries (as well as the cables from corrosion ) while the cars stood idle or in storage ad well as switched to single 12 volt batteries. And while the technology has improved over the years, those are still more compact and, because of the smaller cell size, they do wear out, even with limited use.
After checking the battery and cables, I would look at the mechanical side of the ignition switch and it's connections to the starter before moving on to rebuilding or replacing the whole starter.
I hope that helps you. Good luck and keep me posted on what you find.
Safety Fast!
Art Isaacs