I'm on the fence here, if it's not finished compost and it was left cold, inactive and wet throughout the winter it could contain some diseases from the anaerobic bacteria that would be present. "Firing 'er back up" with a couple active aerobic cycles for a couple weeks should clean it out and get it ready for the garden again though. In this case I'd err on "just to be safe" side...

I've been jaded from a bad experience with anaerobic compost once before so I might be a little overcautious...

You can use that compost as a mulch, but unless it is finished (has no raw organic matter in it) working it into your soil could result in poor plant growth because your Soil Food Web will be busy digesting that unfinished compost instead of feeding the plants you want to grow.

I have some heavy tarps and old carpeting the neighbors pitched.

Since I have seven open mesh bins, I don't worry too much about any one bin. But if a heavy rain is expected:

1) Pile dry -- may not cover
2) Pile a little dry -- may put a piece of carpeting on to to absorb some of the water.
3) Pile OK or in danger of being too wet -- put a tarp over it.

It has been about two years since a pile got really wet.
Since I put the problematic ingredients in the core of a hot pile there is not much concern about odors or pests.

I have a similarly wet Mt. Rotto, but I've found that if I put some compost in a cart with rolled up dry newspapers, the papers do a good job of wicking out the excess moisture when left overnight. Then the stuff is forkable, and I can easily pick out unrotted broccoli stumps, etc.

A friend swears by concrete block composting enclosures simply because they DO retain moisture so well. An asset in a dry summer, but not so much fun in spring when you want to use the stuff and it's still gloppy.

Of course, you can always just do nothing and wait. In a month or so our composts will look at feel much better thanks to the help of microorganisms that work in warmer weather.

Here is a link that might be useful: my website

I build my bins with 1/2" and 1" sticks (the kind that fall out of your trees) on the bottom of the pile for aeration. I have found that even heavy rains don't matter because the water seems to drain out through the sticks. The sticks act like a colander I guess. When I turnover the pile, I try not to disturb the sticks. I was out there turning over the piles today and it all looks very nice, not gloppy at all and we have had a wet winter. Some of those sticks are partially rotted so it will be time to start anew soon.