You need to know your Part A Medicare effective date. That's when HSA contributions must stop.
If you apply to Medicare during your normal window (3 months before or after your 65th birthday), Medicare will be effective on the first day of the month in which you turn 65. If you sign up after your 65th birthday, coverage will start 6 months before you apply, except that coverage can never start before your 65th birthday month.
I don't quite understand how you can apply in May and request a July 1 start date, unless July is your birthday month. In that case there is no lookback period because the coverage can't start before your birthday month. But if you delay signing up for Part A until you retire at age 67 (for example), Part A will be retroactive to 6 months before you apply or 6 months before you apply for social security benefits.
