[RandomOpinion]

One option might be to take a training course for the PMP certification exam and the PMP exam itself, if you meet the qualifications. While the PMP has its detractors (and not without justification), it does enable you to "sling the lingo", so to speak, and also shows you're committed to making the change.

You're probably also going to need to shift your role at your current job so that you're overseeing projects, whether just as a senior engineer doing technical leadership or as an actual manager of people, instead of just coding. (Do not, however, ignore your coding skills; anecdotally, most software PM positions are for technical PMs and may still require a demonstration of coding skills. Non-technical PMs have a much harder time finding jobs.)