As far as I know (personal experience), Dominican employers don't give a care whether you are a legal resident, not even the big ones. You will just start working illegally.
By the same token, they won't pay for your application. You cannot do it without a lawyer, since you don't have a guarantor (they don't care about your job, you need a Dominican national with money in the bank), and the lawyer makes up a guarantor.
You could arrange a marriage for the purposes of getting residency and then file yourself, but I would advise extremely strongly against it.
Lawyer will charge you 500-600 US$. It doesn't matter how long it take since you get a receipt.
Cheers.
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