I have access to a lot of free (and new) 350W PSUs. Is there a way to combine their outputs so it performs like a higher powered PSU? I read about Add2PSU, would something like that work?
You must not join the power supply connectors together as they will compete with each other to maintain different voltage levels.
Use one PSU to power some things, use the other to power the rest.
Taking care to observe maximum power draw for what is connected to each PSU.
A PSU can regulate only its own voltages, so when two PSUs are connected together, they fight one another in trying to regulate the voltages, and they'll aim for slightly different voltages because no two voltage references are exactly the same.
If you want to use two separate PSUs, connect only their grounds and Power-On lines together, and connect one PSU to power the motherboard and disks and the other PSU for the graphics card(s).