Ouch

Hypoallergenic doesn't mean a thing. My fiancee's patents have a fancy hypoallergenic dog that always sets off my allergies

I thought you meant toys like dolls or action figures. Not the actual toys you have.

Ouch, my sympathies. That said, why not do a spot test on your arm first? Thoughts for next time maybe.

Warm water and soap works just fine.

Best to just use a soap you aren't allergic to, warm water, and make sure it is completely dry before storing. Until the hives go away: Benadryl, and if it gets bad enough call your GYN and see if they can call you something in to treat an allergic reaction in that region. Oatmeal baths too, maybe?

Alternatively, depending on the kind of toy in question, some can be put through the "sanitize" setting on washing machines. Obviously, one should check, but it's there as an option. That way, you would only have to dry and store. I imagine you would want to to a rinse before doing actual dishes though. If you were using any, it would be a good idea to test any lubricant used at the same time as the cleaning solution was, just to rule that out. Some people are allergic to even the surgical grade lubricant, and if that's the case it'd be ideal to know sooner so it can get on file.