This Delta faucet is supposed to swivel on the joint (I believe) at the base of the faucet. However the joint seems to be too stiff and it's caused the base of the unit to detach from the counter and now the whole unit swivels. How do I loosen this joint so that it can swivel freely? (Before I tighten the whole unit back on the counter.)
You need to remove the pull-out spout and let the hose drop down through the faucet body, remove the lever handle, then unscrew the cap and remove the valve components, then pull directly upwards to remove the spout.
The spout glides on two o-rings; you need to get a small tube of plumber's grease and grease the heck out of the o-rings, then reassemble.
I could go through the disassembly/reassembly steps in detail but it would be a very lengthy process so maybe just watch an internet video on it.