Hello ExPat, and thank you for writing. Ask your lender to verify your former employment with a VOE -- a Verification of Employment form. The employer should list your termination date as August, because that's when you resigned.
Most lenders will require an explanation for a six month gap in employment. However, your explanation is completely reasonable, and I see no reason why this would prevent you from getting approved for a home loan.
Check out this article on offer letters for home loans. https://themortgagereports.com/8812/offer-letter-mortgage-income-approval
Good luck, and welcome to the US :)