There are a couple ways you can go about getting more specific responses. First, as you mention, you can propound a second set (not amended RFAs). Second, if your requests were specific enough, but the responding party served objections and other responses in an effort to evade responding with "admit" or "deny," you might be able to do a "meet and confer" effort to see if you can get better responses. Then, if they don't provide better responses, you could bring a motion to compel further responses. (Strict timelines apply for this procedure. See Code of Civil Procedure Section 2033.290 and surrounding sections for more information.)
If you are a non-attorney, and you drafted the requests for admissions yourself, your best bet is probably to send a more specific second set. It will save you the effort and cost of filing a motion with the court, and you'll likely get responses faster.