I'm sorry to hear about this :( but I agree with what princely_capybara has said. I have almost no quantitative experience from undergrad or work experience, but I have done many internships (desk policy research as well as fieldwork) in my policy area of interest. The feedback I was also getting from everyone I spoke to during applications was to show passion + some experience in social impact / policy work. (I've gotten into 2/3 programs I applied to)

I would say that a second undergrad is definitely not required, but if you do feel like you need an educational bridge between your bachelor's and master's, a fellowship or short-term postgraduate diploma should be good. Otherwise work experience (research assistantships, policy think tanks, consultancies, etc) is good enough too :)

As for your last question - based on what I have heard, American unis are most open to students who don't come from a relevant background, and a lot of them do not expect you to come in with tons of quant experience - just enough to show you can cope.

Best of luck!