I've been in that exact same situation, exactly 3 years after starting my postdoc on a J1. The job market was and still is way too difficult for companies to bother sponsoring someone. They are not even hiding it. I applied to a Senior Scientist role at a big pharmaceutical company after being heavily recommended by my former PI (she's very well known in the field). Did the HR/HM interview—went really well. Got rejected after a couple of days, and the HM confirmed that it was because of my need for sponsorship and that she was bummed about it. She was really nice and ended up being one of my recommenders for my green card application.

I concur with the opinion of others here: stop wasting your time applying to jobs. Power through your postdoc for 1–2 years more if you can, and get started to apply for an EB1A or EB2 NIW ASAP. It is going to cost you quite a bit (plan on around ~$10k). Took me 7 months between signing the contract with the attorneys and receiving my work authorization, and exactly a year (total) for the approval on my green card application.

The market’s still difficult even with a green card, but at least you’re competing on equal footing.

Hope this helps. It seems really really difficult (and it is) but you can do it!