As long as your wife is working full-time, you can stay on her health insurance without taking Part B of Medicare (the part that costs money, and covers outpatient care), and then pick up Part B when she retires without a delay in coverage, or a penalty.
Since evidently she's paying $125.00 a month for you that's very close to the $129.00 a month Part B costs this year, but who knows about next year. So whether you switched could depend on the quality of her insurance coverage.
You're going to want Part A of Medicare since it's free for almost everybody. That covers hospitalizations.
You don't say whether you're already getting Social Security benefits. If you are, Medicare enrollment is automatic, and you can refuse Part B if you want. If you're not getting benefits, you'll want to make an appointment with your local SSA office to sign up for Medicare, and they can be very helpful, although they're not allowed to advise you as to what you should do.
The states I'm familiar with have an agency called SHIIP that can help, and since they're not trying to sell you anything I can recommend them. The one in NC is excellent.
The one thing I can advise you to not do is take advice from internet forums, although it's your business as to what you actually do. Heck, maybe you can go on the Medicare site and ask about Triple Locks, what do I know (less every day, it seems)?