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Vrtra has mentioned that those bodies are just vessels for his eye. They get switched out every few years so people don't get suspicious as to why the Dragon's attendant doesn't seem to age. Granted certain revelations make the secrecy no longer relevant but, habits are a bit hard to change.
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As for dragons eating in general, they do hunt a lot (see moogle tribe questline and dragoon questline) but some are also known to have consumed plant matter as well from what I have seen (aiatar is said to be feeding on the poisonous matter while resting in the longstop, most of which is plants)
Varshahn can likely eat like normal otherwise it would rouse suspicion. Plus it would add longevity to each vessel so it would stay "alive" even when Vrtra is not using it.