Other local options I don't think I've seen
1. Fairfax County fair, the first weekend in August every year. It includes exhibits from local kids in 4-H. Here is the info from this last summer:
https://www.fairfaxcounty.gov/parks/frying-pan-park/4-h-fair
2. Oxon Hill Farm: Run by the park service, so will likely need to wait until the government re-opens:
https://www.nps.gov/oxhi/index.htm
3. Mt. Vernon (private, so not affected by the shutdown). They have a lot of animals on the estate, to recreate what it was like in George Washington's era. Every year at Christmas time, they bring in a camel as well. Check their calendar since they have certain times of the year with animal centric events too (like sheep shearing in the spring)