I applied to be a non-degree graduate, but I meant to apply to be a post-bac/regular graduate student. Will my first paid application fee transfer over towards a new app?

Unfortunately the $25 application fee you paid for your non-degree application was for the processing of that application. To apply as a post-bac or degree seeking student you will need to pay the non-refundable application fee for that application type and you need to submit any documents required in support of the new application.