Sounds like mildew on the A/C coil. Try spaying the coil with a VERY mild bleach solution. it will smell of chlorine for a few hours but should cure the "wet socks" smell temporarily. A permanent solution will require more work (eg better coil drainage if that's a problem, UV light in the coil, etc)

Thanks skin. But how could it be mildew. It is, literally, a brand-new system. And it wasn't used this summer. The first coil that was put in had a split seam, and that really made the house stink (presumably freon) not to mention causing the AC to not work. But that was replaced.

How could mildew be an issue from the first time a brand new system is turned on?

Why haven't you called your installing dealer?

That's your starting place. Should be easy to locate the problem, solving may be another issue.

IMO

Well, if it is totally unused then as you say, it's probably not the coil (I thought it was a new installation, maybe a couple of months into service). What you might be smelling is basement air being sucked into a poorly sealed return. This usually pairs with an inadequate overall return air flow. As TD says, call your installing dealer and have him fix it!

By the way, R22 and R410a don't have a strong smell (they are technically odorless but the oils used along with some other additives can have a faint odor) so you may have had the same problem back then. Can you describe the smell any better?

I'm going to take a different approach:

When the AC is on, the fan speed and CFM is much higher than when the heat is on.

I'm going to bet that you have either supply side, and/or return side leak in your ductwork. They probably aren't big leaks, but when the blower is on high (with the ac) it's leaks enough to cause air exchange with the basement / crawl space.

Sealing your ductwork will solve the problem AND get more of your heat / AC upstairs in the living space.