I can't imagine it would be hard to find a laundromat--just ask at the desk of where you're staying. You'd be surprised, I've stayed in hotels that cater to business travelers that DO have laundry facilities---Motel 6 is likely one of those.
I live in an apartment building and do my wash in front loading machines twice a week. I just run a cold wash with Oxyclean or even just without anything, then I run a hot wash, and dry what will go in the machine.. and hang the rest over the back of my sofa or dining chairs. I think you can do it! I am willing to bet that after spending so long in the car with your baby... you'll enjoy the refuge of the laundromat for the two hours it will take to get your wash done!
Good luck on your move!