The regular way escrow for insurance works is that for 12 months the escrow company collects money from you, then after 12 months of payments they send the money to the insurance company for the next one year policy period. During the 12 months they are collecting money from you they are guessing what the premium will be. Once a year they recalculate your monthly escrow for insurance and taxes to adjust for any bad guesses.
Based on your question:
Back in May the escrow company sent money to company A for your new term. That was the money ($416.67) they were collecting every month from you for the previous 12 months. In July you found a new company, for lets say $3,600 a year [note: I am making up this number]. You bought a policy from them at a lower rate. You paid company B from your bank account, and cancelled the policy with company A. Company sent you a refund of about 10 months of coverage.
assuming this is what happened.
I will definitely have deficit in my escrow because of the $5K draft
which I can pay back right away using the check.
There is no deficit in the account. After they sent the $5,000 to company A the escrow company continued collecting $416.67 every month to pay the bill next May.
My mortgage company does the escrow adjustment next March. The deficit
is broken down into 12 installments from 03/2024. There's no interest
charge on escrow as I understand which essentially I get a interest
free loan for the next 18 months on that $5K. I'm thinking using that
$5K check to pay the principal. Is that a good idea?
This sounds like you didn't pay for the new policy. Because if you did you would have check from company A for about $4,200 and you would have written a check for $3,600 to the new company. You would have about $600 net after the transactions. You can anything you want with this money. It is yours.
Don't forget to tell the escrow company about the new policy otherwise they might try and send the money to the old insurance company.
Also the mortgage owner will be concerned if you don't have a policy. The old insurance company might have to tell them.
from one of your comments:
I didn't mention it. I paid the new insurance through the escrow as
well. So my lender is aware of it.
I have never been able to tell the escrow company to pay an out of cycle payment. While it is your money, they are there to protect the investors. Paying more money then they are obligated to pay (for the 2nd policy) puts the investors money at risk if there balance goes negative.
So based on the fact that both payments were paid with funds from escrow, you should return the funds. If you don't return it to the escrow account, expect a big increase when they do an adjustment in March 2024.