Yes, there is a workout room at the HHI resort. There are two treadmills, an eliptical, a stationary bike, and a weight bench. (I think that's all...)
At this time of the year, the weather should be PERFECT for walking outside!! Take advantage of all the bike paths. Walk to the beach house and/or walk along the beach. (The ocean might still be a bit cold for swimming, but it will be beautiful there!) Rent a bike at the resort and check out Sea Pines or some of the other plantations. You can go forever on the bike paths there!!
The pools at the resort and at the beach house are heated, so you can swim some calories away as well.
As far as your eating, this is what I've done in the past when I've been on vacation. (Thank goodness for DVC, so I have the kitchen facilities!) I usually eat a healthy, but light breakfast, like cereal or yogurt. Same with lunch -- a light salad or sandwich. Then a "regular" dinner, either at the resort or at a restaurant, but I try to make good choices, without feeling like I'm depriving myself.
Let me know how you do. I'm going to HH in April and hope I can follow my own advice, LOL!!