From what I understand after you are F1 and meet Substantial Presence Test you are filing as a Resident. 

I'm wondering if the employer was supposed to withhold FICA. If not, then how does tax filer pay their FICA dues as a resident? What form(s) do I need to fill out?

If there are more questions to answer to make the appropriate decision please let me know. I would like to learn how to handle this in the future.