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Just installed a new engine. have a new starter, starter solenoid, battery, and neutral safety switch. Just got the battery this week after it wouldn't start. Started up once and wouldn't start again after that. Solenoid would click but wouldn't crank at all. Replaced the solenoid, same story, started once and never again, just clicked. Then replaced the neutral safety switch and it happened again. The whole time the battery has been reading low, even though it's brand new. Today it had barely any charge in it somehow. Haven't checked the exact voltage on it, but it's definitely much less than it should be. Alternator is only a few months old, so i highly doubt it to be that. Right now, the solenoid doesn't even click unless I bypass it with jumper cables.
As I read your post it seems obvious you have a low battery. Have you tried a jump start with a known good battery? The clicking solenoid simply means there is enough power to work the solenoid but not enough to start the engine, not that the solenoid is bad.
Sounds like a parasitic battery drain. You probably have a small spark when disconnecting and then connecting the negative cable, too. Check battery voltage and note it. Unplug the negative battery terminal and put it aside. After a spell, check battery voltage again...it should read an increase.
To be sure: With the negative terminal still disconnected, put a test light between the negative cable and battery negative post. If it lights you have a drain. Start pulling some fuses and note when the light goes out or stays on. If the light stays on when a certain fuse is pulled, check that circuit. Make note of which fuse you pull and what happens.
It could be the alternator has a shorted diode in it, too. Could have happened when re-installing a battery.