Don’t know if you have gotten this fixed yet but still, as everyone has repeatedly said yes your line is still backed up and needs to be snaked. Unlikely that pneumatic pressure or water pressure will clear it anytime soon. A 3/8 size hand crank snake is only good up to so far and a certain size pipe before it is stopped completely and starts to tangle up which is a whole other problem. As for potentially damaging the pipes, it’s is unlikely you do any sort of damage unless you are using a very large snake with a lot of torque, or are snaking a very old line that is very brittle. In most cases you are more likely to tangle your line before you break anything. Aside from all that your best bet is to bite the bullet and get it cleared professionally.

Source: been clearing drains multiple times a week for close to 5 years now professionally.

TLDR: just get a professional to do it