Since few years, H-4 can get EAD. There are some restrictions.

https://www.uscis.gov/working-united-states/temporary-workers/employment-authorization-certain-h-4-dependent-spouses

Also there is lawsuit going on for its removal and the case has been pending. Not sure if USCIS has put a hold on it for new ones.

To get H1-B sponsorship as a nurse, you have to go to rural areas or states. This was a trend with many nurses just a few years back. I saw lots of Nepalese go to Oklahoma from DFW area. In major market and states, I guess they have enough.