Why not just trade basses and avoid any problems completely?
Keep in mind any shims and their placement!!!!!!
And will I definitely need it for the new neck?
It depends on the thickness of the heel of the new neck, you may even need a different shim. However, I would try it without the shim first. You'll definitely know if it needs a shim or not afterwards.
Pardon my ignorance, but how will I definitely know? This is all new to me. I'm think a luthier may be a better choice than me by myself. But let's say for now, no luthier. How would I know?
If you can't get the string height you need with the bridge saddles, you'll need to add a shim (i.e. you drop the saddle as low as it can go and the string height is still too high for you).
Seek ye a luthier post haste
I dropped the new neck in place with the shim, and the action is way too high.