I wrote inspect the old calipers, and that requires some clarification:
Check the exterior and where the pads fit for built-up crap that stop them moving. Ensure that the pins are straight, installed correctly and that the ant-rattle clips are good - if they're not and you have vented discs, leave them off. Solid disc clips are available and cheap, so replace them.
Next, check for fluid leaks from the pistons. If they're leaking, then you've found your problem and your next step is clear: throw them away! Do NOT keep knackered parts just in case.
If they're not leaking, pop the pistons out(now you know what the conical adaptor that comes with hand and foot pumps is for) and check them, the bores and seals. Damage/wear will be obvious.
This will take about five minutes and will inform you if you've fixed the problem or if you need to move further up the system. It's what you would do if you didn't already have new calipers, as randomly buying and fitting parts is a shitty way to fix problems.