383 won't start. It cranks, I have spark & fuel - could it be the battery

I'm new to this forum - maybe someone can help me out?
I have had problems getting my 67 Cuda to start cold in the past, but now it won't start at all. I thought I fixed the problem once when I adjusted the choke, and she started, but now I can't get her to start at all.
The engine cranks, I have spark, I have fuel - but I was wondering if it could be the battery?
Here's why I am thinking this: She sat all winter and I thought I would try to get her going, but the battery was almost dead - she cranked for about 30 seconds and then just click, click, click, from a dead/low battery. I charged the battery overnight, and tried again this morning. I tried cranking on & off for about 10 minutes total while I tried adjusting the carb, choke, etc. and the battery was dead already! It only lasted about 10 minutes!
So here's my question: Even though the battery (when charged) has enough power to turn the engine (with seemingly enough power), could a battery that is old (not enough cold cranking amps) be my problem?
Any advice is appreciated...