It's been once a year for five years...sounds like he'd need a bomb under him to make any changes now. Also, OP, like all spouses in this situation, what you want is for him to WANT to have sex with you, right? And that's not something we can create, sadly.
Your options are:
Break up
Have an affair
Put up with it
You say that you don't want to leave him or have an affair, so you'll just have to put up with it.
I suppose many people in the past were in similar positions at a time when it was very hard to divorce. I'm guessing that a lot of people had to simply make peace with living the rest of their lives without a sex life and feeling unwanted, and having no touch. It's possible to live like that, of course. Not great, but possible. OP, if you decide to wave goodbye to partnered sex forever, as it seems that that's your decision, I recommend you get yourself a really, really good vibrator.
There is another option...he lets you date and sleep with others. Let me guess, he'd be outraged at that suggestion. Doesn't want you but doesn't want anyone else to have you.
I'd be surprised if he'll agree to medical tests. He sounds happy with the status quo, and who cares about the effect on OP, right?
Also OP, do remember that HE could have an affair and leave you for someone else. Or just leave you, like mind did. Be aware that in putting up with the situation, there's a definite possibility that you are trading your current youth for a prize (a lifelong partnership and a forever-intact family) that might disappear. Have a think about how you'd feel if you decided to accept no sex, and then ten years down the line, it busts up anyway. No family togetherness, no enjoyment of grandchildren together, and you'd have sacrificed your best years.
Putting up with this situation is a big sacrifice. You can't get your younger, healthier years back. It's OK to make that sacrifice, but be aware that you are making it.