I’m not expert, but I’ll throw in my 2 cents. I live on the border of Park View and Petworth and bike to work near 12th & D SW as much as I can, if not daily. My commute is only 4 miles and takes around 25 minutes there and a little more coming home thanks to some hills and more evening traffic. I recommend using 11th St south to Pennsylvania (the former has a bike lane most of the way and is relatively quiet when I’m commuting between 7-8am; the latter has the protected bike lane in the center of the street and dedicated signal lights), then cutting over to 15th St (again with a nice bike lane), which will then lead you around the Tidal Basin across the bridge. If you want to continue the beautiful view a little further, jump onto the Mt. Vernon Trail around Gravelly Point. Once you get closer to the airport, there are signs that direct you toward Crystal City. On the way home, downtown traffic can be a little daunting. I’ve had cars cut me off or be generally oblivious to bikers, but by the time I hit NY Ave, I feel less stressed about the traffic. Buy some lights; I bought some cheap ones (like <$10 total for a front light, back light, and tire light) on Amazon and they are fantastic.
I bike in gym clothes and change in the bathroom at work, nothing special there. When I initially arrive I drop off stuff at my desk to give my body time to cool off before heading to the bathroom. While biking, I haul my work clothes and lunch in a backpack, but other people use saddle bags or their rack for this too.
A good bike will make a huge difference. If you’re trying to start this new lifestyle while a little out of shape (as you said) with a clunker of a bike, you’ll feel like an elephant on a unicycle. I initially had a 1 mile commute and used a cheap old bike from Walmart, which was great for that purpose. However, when my commute increased to 4 miles with a big hill, I literally had to walk my bike up the hill (and I’ve run four half marathons in the last 2 years). After that beating, I invested in a really great recycled bike and it has made all the difference. If you need tips on that, let me know!