I Adopted A Horse And Wasn’t Able To Obtain Whether She Is Registered. How Can I Find Out If She Is?

Unless you know her registered name, and can get the last owner of record to give written notice that the papers are lost, (in a rescue horse, this is very unlikely) then you cannot register her.
If you choose to compete in performance shows that do not require the horse to be registered, it’s no problem. Hunter, Dressage, Eventing, Barrels, Poles, Roping, and Competitive Trail are some I can think of on short notice.
These performance shows go on ability rather than looks, so personally I think they are “better” than the breed shows that are all based at least 50% on looks.