Go to the local thrift store, grab a dingy set of clothes, grow out the beard a little, let the hair get a bit shaggy. No one cares/notices the homeless as a general rule (OK, yeah, there are charities, but I wonder if their members notice them when not at work).

Overnight safety depends on the city. Philly in mid winter is a lot different than Miami in mid summer. Weather concerns are paramount, going where you won't be hassled by the cops while being safe from other drifters would be important.