Hugs, you are not a bad mom. You need to be able to get out and about to do things and see people so that you are not stuck at home, and that does not make you a bad mom, it makes you a normal mom!
Will she sleep in a sling? I often found DD would be more comfortable out and about if I carried her, rather than in a car seat/buggy.
She's still so small so try not to worry too much about the naps and things. Perhaps focus on making sure she gets one good nap a day a home and then while you're out let her catnap a bit. She may need to sleep more often then as the sleep will be less restorative, but if you can then let her catch up a bit later on or the next day it may not impact too much on her overall sleep.
What do you think?