I've had mixed results being able to still feed after running I'll Met by Moonlight. If you're not familiar, if you start I'll Met by Moonlight and have Namira's Ring equipped when Sinding gives you the Cursed Ring of Hircine, you will generally be able to feed without Namira's Ring equipped so long as you never equip the Namira's Ring again. Sometimes, however, it doesn't work. I don't know why; Wish I did!

I did Namira's quests, got the ring, equipped it. Headed to Faulkreath and spoke with Sinding. The cursed ring was automatically equipped as usual. Headed up the road to go after the stagg and took out the bridge archers on the way. When I activated one of the dead archers it went right to the archer's inventory instead of giving me the option to feed or search. Dammit!! I opened my inventory and went to the apparel section. The cursed ring is equipped and Namira's is not. Out of frustration I attempted to equip Namira's and it makes a sound but doesn't mark as equipped because the cursed ring is equip-locked as it's supposed to be. But the sound of Namira's equipping gave me hope. I exited and activated the dead archer again and bingo, the option to feed was there!

I even loaded another character and ran the quests with the same results. Attempting to equip Namira's caused the feed option to function. I don't know if it will always work, but definately something to try. Try to feed right after getting the cursed ring and before it gets removed. If you can't feed, attempt to equip Namira's.