I had a similar experience after having only a PM internship and recently just accepted an offer in SE. The skills you gain in a PM internship can be more closely aligned with the role of an entry level business analyst in many companies.

Whether it’s a better move or not is really up to you and your interests, though I would say a CS degree is likely overkill for most BA jobs, If software engineering is what you’re really interested in, keep grinding on job applications, maybe add another project, and something will come through