Currently married to a US citizen but spouse is away for a few months.
The driver's license will depend on your state's regulations. You can call the Secretary of State/DMV in your state and ask, or have someone do it for you if you don't want to make contact.
As far as the job, until your status is adjusted, you'll show up as undocumented on eVerify and you likely won't have the documentation (hence the term undocumented) that any employer who wants to comply with the law will require. You can't put "Married a citizen" on your I-9 and expect that to fix everything. Find a lawyer, get your status adjusted.
Marc Whipple is an attorney, but may or may not be licensed in any particular jurisdiction and may or may not be familiar with any particular law or regulation relevant to the subject matter or related issues. Nothing in this message is meant to constitute professional legal advice or to establish an attorney/client relationship and its content should not be relied upon by third parties. Consult an attorney licensed in your jurisdiction and familiar with the relevant law before making legal decisions.
In my experience, especially in this part of the COuntry, the Northeast, it is not possible. Anecdotally, I understand that in one or two states, Washington and or North Carolina, it might be possible. However, it just does not seem worth the effort and possible exposure to state fraud charges. I'd advise you to wait until after you0 file and get your EAD and SSN then apply for a Driver's License in the state where you actually reside.
You would need to check with the local DMV to determine if you meet the criteria for a drivers license. As a general rule, most aliens cannot obtain a drivers license if they are not in a valid nonimmigrant status or have a pending adjustment of status. You would be best advised not to seek employment without proper authorization, but working without authorization will not bar adjustment of status through a U.S. citizen spouse.
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