How can I get back to the US legally from the Dominican Republic with an expired green card and expired I-551 stamp?

As a returning resident alien.

I would recommend contacting U.S. Customs at the airport you are planning to fly into and confirming that you are a returning permanent resident with an expired I-551 stamp and green card. You should not have an issue given current CBP policy, however, you may need CBP to confirm this with your airline before leaving the DR.

Use your expired card plus the receipt to re-enter with biometrics notice and your passport.

On March 21, 2021 CBP issued guidance that persons with Expired Permanent Resident Cards: ten-year validity and with an expired I-551 stamp like you, may be boarded without penalty, provided the card was issued with a 10-year expiration date. Of course, good luck explaining this to the airline, so you need to find the CBP directive and show it to the airline.

Not exactly stuck but you may need legal assistance to show that you did not abandon legal permanent residency at the next entry into the United States. I suggest retaining an immigration attorney if and after CBP paroles you into the country. Expiration of physical green card or I-551 stamp does not automatically make you lose residency.

You can try simply to fly to the U.S. and explain why you stayed outside of the U.S. Or you could file an SB-1 application at the U.S. embassy.