As a general comment those with HP are certainly eligible for Medicaid, I have seen people post who got it for that very purpose, not Ukranians but all the same.
And I also know that there are other benefits they are eligible for but have not gone in to details.
The EAD is odd, the way I read it it is free if you apply within certain dates and not otherwise but for the life of me can't think why. They say what happens before 30 September but not after.