I do the Damprid thing during off-season haulout (we have winter here), with a small solar panel outside the canvas cover powering a fan to circulate air past the refillable canisters. That works great in winter here (I usually go all winter without having to refill), but definitely not enough for where you are.
During the season I have a Marinco solar vent in our salon hatch that blows air out, and fresh air comes in through the cracks in the companionway hatches. Works great here (never had any mold), but probably still not enough for you in Florida. However, there are LOTS of reliability problems with Marinco's latest solar vents requiring me to make annual warranty claims.
On our prior boat (C250) I had a small Peltier dehumidifier that sat in the sink, just like you suggest. It consumed 72 watts and ran constantly, but it's very inefficient - it didn't remove much water for the amount of electricity it consumed (maybe a pint a week at most). It was good enough for my little 25 foot boat in Pennsylvania, but probably not enough capacity for you in Florida, or for a larger boat anywhere. The Peltier types are sold under various brand names, the most popular being Eva-Dry. I'm telling you this so you can avoid them, since I doubt it would have the capacity you need.
For your needs, you probably need a full-blown dehumidifier (which runs with coils and Freon, like a refrigerator), which will be larger/heavier and perhaps a little cumbersome to put on your galley counter. You'd get these at Home Depot, Best Buy, or other appliance store. They usually have built-in collection pans that hold a gallon or tow (can go about a week), with a punch-out that you can attach a hose to that could go into your bilge if you need more capacity. That is what I would recommend for summer. Probably would fit under your salon table with hose going to the bilge.
Finally, if you have or want to consider a full-blown 120v marine reverse-cycle air conditioner in your boat (might be nice for Florida), they usually have a "dehumidifier mode" which will do the job much more economically than running the AC full blast. You'd run AC while you're on the boat in the slip, but run dehumidifier mode when you're away from the boat. Might be worth considering in Florida. This would require you to leave a through-hull open for the intake water. Water usually exits above the waterline. My boat has one of these, and it makes a huge difference when we spend time in the slip. I don't need to run dehumidifier when we're away because the solar fan is enough for us up here.