Getting a postdoc is far from "making it." A good portion of postocs are just glorified technician positions exploiting recent PhDs who have not given up on the dream. So being unaffiliated is nothing to be ashamed of. If anything, it shows determination. I know very few people who have published without institutional support, so I think this looks good on you.
Did you do all of the work for this manuscript while being unaffiliated? If you did a substantial portion of this work as a PhD student, at least in my field the custom is to use the affiliation of the PhD institution. If you did the work while not being affiliated, then good for you.
It used to be that people only used official email accounts (e.g. .edu) to submit papers, but it is now very common for corresponding authors to use gmail accounts. Nobody will think a thing about it.
On a psychological note: are the people you suspect will look down at you, are these people cutting you a check every week? If not, the hell with them. You gotta make a living, and there's no shame in that. So you keep putting postdoc applications out there, keep publishing, and you'll land your job.