Similar setup to you - outdoor storage, shaded, afternoon sun, mooring cover. I have none of the issues you have, but there are some differences between our use and yours. You have the snap-in carpet, your area is way more shaded than mine, and your boat is 2 years older than mine.
We almost immediately got rid of the carpet just because it was hot as snot underfoot. When we did have it, whenever we removed it, there was always SOME moisture underneath. My suggestion would be to start pulling your carpet after a good weekend of airing it out, and see if that helps. Going to SeaDek, we've NEVER Had an issue, but you may not want to drop the cash if you're going to a bigger boat. Make the move either way - SeaDek or bigger boat. Realize that with a bigger boat, you're STILL going to have to deal with this, so you might want to hold off on the boat and test out SeaDek on the current boat so the wife gets used to it, and it's likely it's going to be HER decision to upgrade to the SeaDek on the new boat - win for you!
If you've never done anything to your cover since you got it, that's likely part of the issue. Your old canvas cover was at least somewhat breathable, your black mooring cover is not nearly as breathable, and it's strapped down tight to the boat. It's supposed to be water "resistant", and IIRC, there's some sort of liner or coating on the underside to help it shed water. If you've not touched it, the water shedding coating is likely worn and needs a refresh. Mine was starting to let greater amounts of water in, so I used 303 Fabric Guard on it, and since that point, it's been perfect at shedding water. Seams are a must to really coat, I've got a post somewhere here where I did mine, and that'd be my second suggestion for things to do.
The last thing that we do is that at the very least, we take a towel to everything inside prior to covering it. You say you're getting water in the gutters by the seats, which to me, points to water intrusion, either by the tower or through the cover, or both. Nothing you can do about the tower area, but the cover can be addressed. Fabric guard is a cheap addition and you can likely extend the current cover at least a season or more than it would last you. You'll also get an idea of how well it works, and you'll likely want to keep doing it on a new cover when you DO decide to make that move. Drying everything out before putting the cover on will definitely help you identify if the cover needs attention, and also help with some of the issues you're having.
Are you having issues with water in any of the storage areas? If those are getting wet as well, that's not helping your situation, so mebbe check that also.
As long as you have power, I'd drop a fan in there as well, because it sounds like you have a lot of excess moisture in there, and that might take a bit of mechanical aid to truly dry things out. Extension cord and a timer on the fan seems like it would help. With night time temps dropping, you may also have dew/moisture buildup overnight that's never really getting a chance to dry out, and is contributing to the problem.
TL;DR
Dry your boat after use - free
Pull your carpets - free
Fan inside under the cover - cheap/easy/maybe necessary
303 Fabric Guard the cover - cheap/easy/can't hurt to try/maybe necessary
My current situation (and none of the issues you are having):
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