Hello epartoyan,
Thanks for reaching out to us in Apple Support Communities about not seeing your printer status in the Dock when printing. We're happy to help by looking into this more!
While the following resource is for general printing issues, it provides some relevant troubleshooting steps to try:
Solve printing problems on Mac
If you try all these steps and still have issues, let's have you set up a new user and see if the issue continues under that new user:
Set up users, guests, and groups on Mac
Hope this helps.
Best Regards.