General caveat that this is just my opinion and there are many ways to approach this. My view: I would focus far less on building projects and developing on-the-job skills prior to getting a job and far more on improving your interview skills. I'd do that through a combination of leetcode problems, mock interviews, and (most importantly), actual interviews. Without a CS degree or job experience as a software engineer, many companies just won't consider you regardless of how many side projects you have or technologies you can list in the "skills" section of your resume. The companies that will consider you, what will really matter IMO is how you actually perform in the interview. For better or worse, the skills that are necessary to be successful in tech interviews are generally different from on-the-job skills.

I would just apply to tons of companies (e.g. try to send out five applications a day, or something along those lines). Your skills as an interviewee will improve as you do interviews, until finally you get an offer you like :). Also, this is often a question of getting one offer, and then telling all the other companies you've applied to that you have an offer with a time limit to accelerate their processes. FOMO is real :)

All of this is predicated on being able to get interviews -- I'd expect a low hit rate, but if you're getting zero interviews, you'll need to adjust your approach and either improve your resume/cover letter or try a more personalized approach, like finding opportunities through your network or meetups, etc.

Re: jobs, I would apply to entry level positions. I would not apply to internships. I think your probability of getting internships vs. entry-level jobs is only slightly higher, so just go for entry-level. With entry level jobs, you're still going to be hired for your potential (not experience), which good companies recognize.

As for your resume, definitely keep it to one page.