If you sign the driver of Gally shuttle rarely asks for ID or visito pass. I guess it's not wise to expect that though. If you Metro to NY ave, you can walk on Florida Ave or walk on M street. You say you will be bringing sthings with you. Not a good idea to walk in that area with your stuff. You could get mugged.
Suggest go to NY Ave- Gally station and try to use shuttle. You might get lucky. If not, then walk.
Gally's shuttle goes to Union Station so that is why it was suggested, that's all.
The temp has been in 40's lately, much warmer than normal.