Should alternator be putting out 13-14 volts?

... I know the nightmares that can happen with these Ammeters and their wires. That's why I replaced everything.
Not a problem in your car. In 1963 & 1965, they ran the thick ALT & BAT cables thru special "buss lugs", not the 56 terminals (spades) of the other wires. So, no melting problems. I even put a 1965 firewall connector in my 1964 to get that feature.

You can get an electronic Vreg for your car for ~$11 (rockauto). If still the old "mechanical regulator", you need to periodically open it and clean the contacts and adjust (read manual). Smart to lose your old points ignition and upgrade to electronic. Many conversion kits. I put a later e-distributor in my 64 slant, driving a GM 8-pin HEI module & coil. If a small block, the $40 Chinese distributor on ebay is tempting.