LORs sound good, and yes, definitely try to get that independent study in if you can. That's how my research started. I was originally only going to do it for a semester, but a year later, I'm still at it
One thing I regret is not having done a thesis in undergrad, because you need a writing sample when you apply. That was kind of a headache for me, as I had to scramble and put together different papers, instead of having one coherent one.
For me, I was originally going to go the master's route---my goals are think tank or government as well... however I did want that PhD. I didn't really think I had what it took to get accepted to a PhD program right out of undergrad, but my advisor urged to try anyway. What I ended up doing was applying mostly to PhDs with the option of being considered for the Master's if I didn't get accepted at that school (some schools have this option, some don't).
I think it's good to have other people give their opinions on your situation, but at the end of the day, you know yourself best. Do research on schools that you think would be a good fit---both Master's and PhDs (check out GRE scores, GPA requirements etc). Some Masters are designed for careers, and others prepare you for PhDs. Ask your political science professors as well, as they've been part of this "world" for awhile I presume.
Sorry for the long response, but your situation sounds eerily similar to what mine was a year ago. Best of luck.