Faulty battery, bad alternator, or both (photos)

At least those melted connectors are easy to find and replace. They are termed "56 terminal" or "Packard 56" and can be bought on ebay or on-line auto electric sites. Re-wire w/ same color & gage wire. No sense fooling with an old harness since most 40-yr old underhood wires are degraded. I totally re-wired my 64 & 65 engine bays. Find the short to gnd before powering. You need a multimeter w/ ohmeter (free at Harbor Freight).

Re alternator. Easiest way to prove it works is to disconnect the factory field wires, jumper 12 V direct from BATT+ to one terminal (doesn't matter which, blue wire is factory) and jumper other terminal to ground. That applies full field current. The alternator should now give max output, which you will see on the dash ammeter. All that current goes thru the bulkhead connector (thick red & blk wires). Those often melt due to corrosion. Before that, verify that both field terminals are isolated from ground (ohmmeter), since a field short in the alternator could be why your IGN1 wires fried.