Typical with all cars. You have an apparent battery connection issue usually.

When a heavy electrical load is demanded like trying to start. The connection goes open. Best to start by really cleaning the battery posts and battery cable clamps. As for myself I get a little more anal.

I take a twelve volt test light with say the headlights turned on but not illuminating. You have to have a potential electrical load in place for the testing to work.

Touch the center of each battery post with one probe and the battery clamp with the other. You can do this with a voltmeter as well. I just keep going until the test light lights. Or a meter shows battery voltage. This indicates where the electrical circuit is open.

I also treat connections to make them more reliable. Basically keeping them from oxidation. One has to keep in mind that these cars and many of the electrical connections are really old now. Since your battery is new and it is reasonable to assume the posts are clean. Clean the inside of the battery clamps. This is a good starting point.

Your old battery may have still been good. Unless by experience you know it is bad. Always have it tested free at one of those auto parts places that do this. Before buying a replacement. Your description in general does not indicate a short.