Quality over quantity. Apply to a few jobs, apply directly, really get to know the company before you apply and show that in your application by completely tailoring each one to the job.

I'd also market yourself as a Software Engineer (or a better known job title) rather than a "Product Designer". In the right company, an engineering role will see you dip your toes into all aspects of the product development process, which seems to be what you're after.

Lastly, remember that you need to stand out at every step of the recruitment process, from the easily discarded stack of resumes, to the checking your social media/github profiles, to the interview and technical assessment.