Help understanding refinancing after vehicle purchase and extended warranty cancellation/refund

On May 30th, I bought a used 2015 Mazda CX-5 GT with about 100k miles in IL for about $14000. I came in knowing exactly what loan terms I qualified and wanted (6% APR for 36 months through my local credit union with a $6500 down payment), and told the car salesman this. I also have no debt and a >800 credit score. He printed out a vague paper with a monthly payment that matched my calculations and my requested terms. I go in to sign and pay the down payment with the manager, and he pulls out the "oh this car qualified for a warranty! yada yada yada," flipping papers quickly, and never said that it would cost $3000. So that increased my amount financed from about $9000 to $12000, and gave the same monthly payment at 10ish% APR for 60 months 😭.

I should note that this is my first time buying a car or anything more than the cost of a laptop, I went in alone, and I thought I came in prepared to get the loan on the terms I wanted. But, the manager nor the salesman never detailed out the "this is the APR, this is the length of the loan, this is the monthly payment" and only verbally told me the monthly payment. Of course, everything is on the paperwork that I signed. I've definitely learned my lesson about being sure to read over everything I sign, even if I expect them to do what I requested. After looking into the extended warranty though Ethos Group, their BBB is riddled with complaints about claims never getting reimbursed or delays. I already emailed in cancellation forms according to the terms on my paperwork, and it will refund to my loan in 4-6 weeks (hopefully).

My question is how will my loan change when the $3k is refunded from the $12k financed amount? I think it would reduce my principal balance, but how would it affect the term and the monthly payment? I know I can refinance with my local credit union after 5 months, but am I able to walk into the dealer and ask them to change the lender and loan terms after the warranty is refunded in 4-6ish weeks? I know if I try to cancel the warranty with them they will drag it out until they get their kickback after a certain period, but if I walk in today and tell them I sent in the cancellation for the warranty, can they change the terms now? Any advice or anecdotes would be greatly appreciated!