I suppose I'll just wait until I find a good applicant and be grateful I won't have a vacancy this time next year. 

Not that it cant or wont, but unlikely.If OP applies for homestead on a new property, then I personally would not lie during the application about receiving the exemption on the existing property.

But, with a strong application process, we don’t have many evictions.  

Is your DSCR ratio greater than 1-meaning are you cash flowing (according to the lender's criteria of mortgage, property taxes and insurance (and HOA) if applicable).

When they realized that it was held in an LLC they denied my application.

Navigating the Surge in Rental Application Fraud: A Guide for Landlords and Property ManagersIn recent years, the rental market has experienced a troubling rise in application fraud.

Do you do background/credit score checks on all applicants and have minimums? 

As always, this can be mitigated with proper screening of applicants by verifying rental history and contacting prior landlords about the prospective tenant's past.

Nowadays, people can make paycheck stubs/W2's that look convincing so just make sure you invest in software that gathers verified information on applicants.  

Applications were being lost, or the website wasn't permitting tenants to fill them out.