Decades ago, a friend was bicycling touring. They had all their camping gear along, but the occasional night in a hotel was tempting. They looked for motels that, late in the evening, clearly weren't full (look at the parking situation) and tried to get a bargain room. "We're bicycling camping, but could afford $25 for a room you wouldn't otherwise rent, is that a possibility?" You staying there isn't free to them (hot water, labor for housecleaning, those little bars of soap) so don't think $10 is an possibility. Be friendly, and be neutral about the outcome, whatever it is.
Consider also that many motels now offer a free breakfast. If, from 7 to 9 am, you're at a breakfast table, writing letters, watching national news, and making yourself another waffle every 20 minutes, well, that 's a lot of calories at no added cost. There aren't any veggies to be had, but you can seriously catch up on fruit consumption between the apples, bananas, canned fruit salad, etc.
And strike up conversations, "Hi, I'm hiking from Mexico to Canada this summer. Are you on vacation, too?" People like to help out in defined, specific ways. Mail your drift box when they drive through the next town, gift you a couple of AA batteries, run you over to Walmart for a pair of socks, etc.
Lastly, there is the art of the sponge bath. Ideally done in a single-occupant restroom, but sometimes I'll take a water bottle of warm water into a stall with me. I most often do this in airports after red-eyeing 2 or 3 flights, but gas stations and restaurants work just fine. Try to avoid peak times and leave the restroom cleaner than you found it.