I wouldn't have the expertise or ambition to rebuild an automatic transmission as you already have! Chasing these symptoms unfortunately will likely lead to transmission failure which is probably how your journey began.
Since a good transmission still up-shifts when “cold” what comes to mind is the change of oil viscosity. Thinner oil might allow flow through a partially clogged passageway or one of the valves in the valve body is sticking. This mimics a low fluid level or a clogged filter you're finding which I'm assuming is due to the excessive wear stuck in first gear.
Not having had this issue myself, a quick google search found several discussions on C4 C6 transmissions pointing to the pressure regulator, governor, vacuum diaphragm, linkage, band and/or servo possibilities. Sorry there isn't a single answer out there so it becomes a process of elimination.
In your search for guidance don't limit to Thunderbirds alone as these transmissions were used across the Ford line. You might find the resources you need in one of the other Ford forums. Best of luck and hope to see your Thunderbird on the road soon!
I've included a photo of the Cruise-O-Matic diagnosis guide from the 64 - 65 Service Manual.
[IMG]
[/IMG]