I'm surprised no-one has said this already, but whilst a technical background helps, being able to code is never part of a Product Manager's job.
You need to own the product & it's vision. This means understanding the market, competitors, and most importantly, what users actually want/need so that marketing can target them. In ANY job, you can ALWAYS be talking to users in your spare time.
And the specific answers to your questions:
1. I would try & find a PM role in a smaller startup, as your Marketing skills will also be seen as valuable (i.e. you will essentially do a bit of both roles, but this will give you the PM experience).
2. Definitely do not go back to study. Waste of time/money for something that a degree can't teach you anyway.
You need to own the product & it's vision. This means understanding the market, competitors, and most importantly, what users actually want/need so that marketing can target them. In ANY job, you can ALWAYS be talking to users in your spare time.
And the specific answers to your questions: 1. I would try & find a PM role in a smaller startup, as your Marketing skills will also be seen as valuable (i.e. you will essentially do a bit of both roles, but this will give you the PM experience). 2. Definitely do not go back to study. Waste of time/money for something that a degree can't teach you anyway.