Totally clueless, looking for a second opinion

Hi bikewrench!

My daily commuter bike has recently developed this issue where the pedals turn when I’m pushing the bike forward. There is no issue when riding it, and I haven’t felt any resistance or grinding, there is no noise. When coasting with my feet off the pedals they only occasionally move. I’m really only invested in fixing it because it trips me up when I push the bike. 

As the title says, I’m totally clueless on bikerepairs – I own no tools and I usually take my bike to a shop every 6 months to get it tuned.

When I took the bike to a mechanic with the above issue he immediately said I’d have to get a new rear wheel or a completely new bike – quoting approximately 185 USD for the repair. 
In my uneducated opinion this issue is simply not worth the quoted price, and I have a hard time believing that changing the wheel is the only fix. 

English is not my first language, so trying to google this hasn’t helped me much as I can’t intuit what search terms to use (in my own language there just isn’t as much content, so no luck there). 

Could there be something else going on, which would be cheaper to fix? And if yes, what should I tell the next mechanic, so they don’t just tell me to change the rearwheel again? 

Thank you so much to anyone who has time to reply!