NorthByNorthwest

I was in this exact situation back in 2017 where the company offered to do L-1B with blanket approval that would have had me working within weeks, but I opted to stick with K-1 to AOS faster without any dependency on the company in case my new role didn't work out. However, at the time EAD approval times were 3-4 months so I just took a long-ish vacation and the company was fine with that. With the current wait times I'd probably have gone with the L-1B, I'm not sure my company would have waited a year for me to get EAD. If it had been L-1A on the table it would have a no-brainer since you can transition to EB-1C without PERM certification and have a green card in hand in less than a year anyway.

I'm surprised nobody suggested making sure you have USPS address change with forwarding in place in time for your move. I don't recall if the actual card shipment has some restrictions as to not allow forwarding, but I had no issues with other USCIS mail getting properly forwarded when I moved while waiting for an important I-797. The USCIS address change is of course important, but USPS address change takes care of the gap.

Correct. Not sure how quickly that system updates, worst case it may take a few days, but the site will give you the I94 record number.

Copies of the most recent I94 are typically available here if it was issued electronically: https://i94.cbp.dhs.gov If you are not able to retrieve it at that site you might need to apply for a replacement, see page 10 of the N-600K instructions: https://www.uscis.gov/sites/default/files/document/forms/n-600kinstr.pdf

Plenty of good info specific to NM so far, here's some generic info from an older post that gives you an idea of the worst-case costs involved if they don't qualify for any state benefits: Elderly immigrants can buy into Medicare after being legally present in the US for 5 years, at that point they become eligible to sign up for Medicare for the full premium. This is currently $506/month/person for Medicare Part A and $165/month/person for Part B. They would both need to sign up as soon as they become eligible, or there would be additional monthly penalties if they ever want to sign up later. In most states they will not be eligible for Medicaid until being present for 5 years either, and even then they may not be eligible depending on your income assuming they're part of your household. In summary, after 5 years you'll have to pay at least $1,500/month for basic Medicare for the two of them, and that's before any copays and deductibles. During the first 5 years you would have to buy private insurance that can get very expensive for elderly people. A quick glance at the NM marketplace shows the cheapest plan being around $1,500 / month for an elderly couple born 1940 and 1950 and that's also the cost before deductibles and copays. So lacking any state benefits, assume a minimum of $18,000 / year if they are healthy.