Usually I sell the food I find,but sometimes I post the more valuable things on the Exchange or give it to my bridge officers.Because I don't really see it's use during combat since the large hipo is a more efficient way to heal.
The other use for food,besides tribble breeding,is as quest item for the duty officers' assignments.
Also,you can buy Chateau Picard and Earl Grey tea from the bartenders.
Since you are a science officer you probably don't need hyposprays either. You or your other science officer on the away team can heal your party with medical tricorder or other ability. To help your bank problem I suggest you list whatever food you want on the exchange, recycle the rest and recycle your hyposprays and anything else that you haven't used. Also, don't bother keeping commodities as you can buy most of them when you need them from the replicator. You didn't mention you had any commodities, but I figured I would jump the gun on that one so you wouldn't. If you don't regularly use it, you can safely lose it is my saying! If you decide to become a gold member, you can go back to saving random crap. Otherwise only save the bare minimum. Hope it helps.
Aside from a minor and very temporary damage buff, the only use I have found for food is to do the Culinary DOFF missions you get from your ship's chef in the mess hall on the crew deck of your ship when you visit the bridge. The problem wiith that mission chain is that (1) the only rward for completing is that it "may" open up additional missions, and (2) completion of it requires a supply of Klingon Racht, which seems to be almost impossible to find unless you pay an arm and a leg to buy it on the Exchange. I have just about given up on it because after months of searching with all of my alts, I've only found one Klingon Racht and I need two to complete the mission chain. If I were a Silver Member and had the limited inventory/bank space that Silvers have, I would have sold off all my food long ago.
p.s. If you want to take the time to do it, you can usually find some food items on the Exchange selling for less than their salvage value. Since there's usually a vendor close to the Exchange, you can buy as many of those as are available at bargain prices (anything under 10 ECs) and then sell them at the vendor for a quick profit.