If I charge the battery in another laptop and then put it in this one it will boot up, but only on battery power and until the power goes out on the battery. I have replaced the A/C adapter twice to double check the adapter being bad/good and have had same results with all of them.
The other laptop that you charged your Laptop's battery in, was it the same model of laptop.? And did you use your mains adapter.?
If you answer yes to the second question then the problem is not your battery or mains adapter but the laptop itself. This first question is just curiosity.
If its the laptop there are only 2 things it can be, and only one (might be) worth doing. The cheap option is if the power plug on the laptop has been broken, this is often not visible at all, so in reality its going to cost you money taking it to have it looked at by a computer engineer and then fixed - cost unknown -get a quote to fix it, and use someone that's no-fix no-fee.
If the power connector on the laptop is OK then its the motherboard, at which point you need to get a new laptop as it wont be worth fixing a laptop that old. Laptop motherboards + fitting usually cost 50%+ of the cost of a new laptop which will be much better in every way than your current one.