Let me just talk about the HSA first.
Eligibility is determined by your coverage on the first day of each month. All contributions are counted together (employer, employee by salary deferment, or employee by after tax contributions.) Your eligibility determines the total amount you can contribute, but when you contribute during the year is not important and not counted. Lastly, once you have money in an HSA, you can use it to pay for medical expenses for yourself, your spouse or children, any time you want, even if you aren't eligible to make new contributions.
If you are enrolled in a family HDHP (and assuming you are under age 55) you can contribute $691.66 for each month you are covered by the family HDHP and have no other disqualifying coverage. As soon as the HMO covers you, you are not eligible to make HSA contributions. If we assume you were covered at your employer on 1/1/24, and covered by your wife's employer on 6/1/24, then you were eligible for 5 months and your contribution limit is $691x5 months=$3458.33. When the contribution was made doesn't matter, even if the contribution is made after 6/1/24. What counts is that you don't go over the limit of $3458.33 for 2024.
So if your only contributions are the free money from your employer, that can continue. You can even contribute an additional $958 if you like, via payroll deduction or by a direct contribution that you can take a tax deduction for.
If your total is more than $3458, you have until April 15, 2025 to remove the excess. This is not a regular withdrawal, this is a special procedure you need to discuss with the HSA bank. If the excess contribution earned any money (interest or investments), that income must also be withdrawn and declared as taxable income on your 2024 tax return. There is no penalty if you complete the excess withdrawal on time and using the correct procedure.
I also believe the benefits administrator is incorrect about leaving your spouse's plan. Generally, there is an open enrollment period once per year, where changes may be freely made. Changes outside the open enrollment period can only be made if you have a "life event" such as birth of a child, marriage, divorce, or getting or losing a job. You need to learn when your wife's employer's open enrollment period is, then you may be able to change insurances if you decide that's best.
I'm not going to offer an opinion on which insurance is better, that's very complicated and very personal.
