I'm about to apply to a job where I hand in a physical application in a few days. It is entry level crew at a new Trader Joes they are opening in my town, so it's not some huge position I see making a career out of. It doesn't ask for a resume, but I'm thinking about doing one. Do I even need one at this kind of job?
This is one of those cases that you see in all "Looking for work!" threads where some people will scream you need to suit up no matter what kind of job it is you're interviewing for and some people will say to dress a little better than their average employees.
I used to a do A LOT of hiring of people into jobs a lot like your Trader Joes job, and the resume never hurt anybody if it was well written. If it was bad (typos, bad grammar, etc.), it killed the application for me. You seem like a smart guy, so I am sure you will put together a good resume and it could very well help you.
ON THE OTHER HAND, I knew people with my job at my old company who also did a lot of hiring and they found people bringing in resumes or wearing suits to lower level jobs to be obnoxious and they'd toss the app or never seriously consider the person. Yes, I know, THEN I WOULDN'T WANT TO WORK AT A PLACE LIKE THAT THEN and all that, but the point of this is trying to get a job.
If I were you, and again, I have hired literally hundreds of people just like you, I would have the resume. If not a resume, a BRIEF cover letter on your app explaining your gaps in work. Your work history is sketchy, and that is the #1 thing people look for in jobs like this. If you don't cover the work history issues, you probably won't get a call back. Also, in a situation like this? Phone calls aren't so bad. I know I hired quite a few people because they called. At least they show they are willing to make an effort. Calling and pestering doesn't work for upper level stuff IMO, but it will work here. They need to hire a ton of people. They want people who want to be there and won't be wasting their time.
Be one of those people. Good luck!