Go car free. I moved here 7 years ago, and lasted 3 weeks with my car in Adams Morgan. I definitely do not miss it and can’t imagine having the hassle of the car now! (similar to you – i had to put $700 into it a few weeks before the move).
I get cold very easily, but I do not have trouble with winters here, as long as you are not waiting for the bus for too long. If you are walking, you are keeping yourself warm.
My bf and I have bikes, capital bike share for emergencies, and zipcar. But honesly I walk or use the bus most of the time, I use the metro only a few times per month. We do rent cars from Bethesda, Union Station or National airport for weekend trips sometimes. And we just use car insurance on our credit cards.
One more plug for zipcar – it’s amazing to have options to rent different cars vs being stuck with the one you have. I have rented pick up trucks for my craigslist furniture purchases, bigger cars for Ikea trips to fit everything inside, and your regular 2 doors for trips to the burbs.