Well, if you want to train for gymanstics, and especially want to do ring work, you need to definitely hop on the rings more often than 1-2 times a week. Gymnasts bodies are byproducts of their training, so therefore, adding "excess mass" (fat, muscle, whatever) disturbs their strength/weight ratio (extremely important, especially on rings!). So I'd say keep lifting, but do more gymnastics (if that's what your ultimate interest is) instead of lifting (3 times a week for gymnastics supplemented with 1 or 2 days of weight lifting in the strength/max effort kind of range or whatever works for you, but if gymnastics is your interest, supplement it with heavy weight lifting).
To be honest, I've never heard of a gymnast having a phenomenal diet. If anything, they're probably some of the few olympians who have crap diets and still are strong as hell and have some of the most impressive physiques.
Lastly, you need to take care of your shin splints first before you decide to do floor work or anything involving high impact on the legs (landing a back tuck with shin splints must suck!).
Also, at 5'8", you're going to have a bitch of a time on rings with things like iron crosses, planches, and malteses (your disadvantaged leverage position will be even worse than a shorter person.) So definitely work rings a lot if you want to do them!