Sounds good - you are in no rush and getting armed with knowledge prior to a possible buy. I would suggest buying a chassis manual which could help you get familiar with the car.
The 347CI and 4-speed hydro was a good pulling combination - plenty enough power and zippy that it was a fun car with the transmissions gear ratios. As I recall on the '56 X-member, it was essentially a thin sheet metal folded over to create it and thus the water/moisture could get in there and cause the rust/rot. Anything from up "north," or even near the coastline (salt air), has to be carefully inspected because of the salt used to melt the snow/ice in winter time. This rotted away cars long before the drivetrain sometimes wore out.
I always liked the '56 4-Dr HT and thought it was "cool" looking with all the windows down and no pillar like a sedan. Ton of room inside a 4-Dr. If you get a 4-Dr HT, inspect the post between the 2 doors which is the support/hinge for the rear door. I had a rusty '57 Caddy that I thought I was going to bring back to life, but was too far gone once I began to really tear into it. Someone had left the rear windows down for years and it rotted the bottom of the doors out although the outside skins looked perfect and the center post also got hit badly with rust and would have been tough to get repaired and cleaned up of all its rust.
Just be patient and hopefully the right one will drop into your lap.