Classic/Antique Car Repair: 63 Ford Falcon Futura convertible V6 170, 63 ford falcon, redone

Question
Bob:
I recently had a 63 ford falcon futura completely restored for my daughter. The engine was redone first and then sat for several years while the body work was being completed. It took approximately 4-5 years and it looks perfect. After a short test drive the engine overheated and we took it into
another shop to have all of the seals and hoses replaced.  However, we have had a few problems that I cannot seem to understand why they are happening.
Problem #1: The battery fails after leaving the car unused for days. It is a new battery and it has happened a few times.
Problem #2: When driving on the freeway; after reaching speeds of over 55, the car seems unstable. Is there something that we can do so that it can safely drive at 65-75 miles per hour?

Answer
First I have a question about the battery. You say that it is new. Was it new and sat around waiting for the restoration to be finished? Let me know the date of purchase. On the second questions I would get the car into a good front end shop for an alignment to start with. Will be waiting to hear from you.
Brad