proper upkeep of your personal hygiene is absolutely important. especially if you don't want the whole world to know you're homeless, smell is a dead giveaway. also it's important for your health, the best way to avoid infections, rashes and becoming sick is to stay clean.
the key is to find public restrooms where you have enough privacy to get yourself clean. many bigger chain gas stations have washrooms for customers, and from what i've seen they're usually the one toilet private washrooms. if you're quick you can strip down, leave your pants hangin around your ankles, and use a washcloth in the sink to get clean. it's also handy to have a packet of moist wipes or personal fresh wipes handy for sweaty times when washing up in the sink isn't an option.
other places to find public washrooms: Wal-Mart, chain coffee shops like Tim Hortons [yes, i live in Canada..] some bigger grocery stores or 24 hour places have public restrooms, laundromats, the library, bus stations, truck stops, some parks in bigger cities, and colleges or universities. if you're young enough or at least look young enough and dress kinda like the students, you can walk right into a college or campus and look like you belong.
cleaning in a sink will keep the stink off for a while, but having a nice hot shower is still the best way to go. even if friends or family don't have room to let you stay, they probably wouldn't mind you using their shower every once in a while.
i always have my soap, cloth, a pack towel and a toothbrush in my bag in the event that a shower opportunity will arise. getting a gym membership, specially one you can use in multiple cities could be a good investment. this will give you access to not only showers but the pool, sauna and yeah, the equipment. it could also serve as a good place to hang out if it's super hot outside, or cold and rainy. if you happen to live near a large body of water with public beaches, there's always change rooms and shower rooms available to the public, and as far as i can tell they're always free.