4 week long waiting list for STD testing in my area, how can I get tested sooner!

About a month ago my partner's condom broke while we were having sex. I am concerned that I may have an STD. I have been trying to make an appointment for STD testing but the soonest I can get an appointment is 4 weeks away. I have called every gyn on my insurance list and all the free clinics within a reasonable driving distance. I have had them put me on their call lists in case of cancellations but it I go to school and works so it is unlikely that I will be able to get away at a moments notice. I looked on the internet for at home test kits but they are $200-300 dollars and I am not sure how accurate they are. I am afraid if I wait the 4 weeks to get tested and then the additional week for the results to come back that there may be permanent damage done. What should I do?