How to avoid getting screwed during hospital stay?

I live in Texas and have Cigna, 10% coinsurance with a max $5000 out of pocket for our family (we’re all on the same plan). I had my first child 2 years ago and got so many scammy bills – bills that were copies of ones paid by our insurance that the provider was trying to double bill for, balance bills from the anesthesiologist where he tried to bill us what the insurance wouldn’t cover (why does no anesthesiologist accept insurance?) and now bills from 2 years ago that were never issued before that we have to fight under Texas timely billing law.

I’m going to give birth again (different provider and hospital but same insurance) in May and the scammy bills are worse than the stress of postpartum. Is there anything I can do ahead of time to limit my exposure to unexpected out of pocket costs, balance bills, and out of network doctors? My last birth was an emergency c section and I couldn’t exactly verify the insurance coverage for the anesthesiologist in the 2 minutes they had to get the baby out. Any advice would be appreciated!