Check out Hystersisters.com for all things hysterectomy. It is extremely rare (I used to moderate over there) to do it for all the reasons stated above. Having a hysterectomy, even if they do keep ovaries, can lead to menopausal symptoms, as the ovaries are somewhat "shocked" from the interruption in blood supply. And surgical menopause with both ovaries out is NOT the same as "natural" menopause. Especially at your age, the bone loss and other issues (such as increased cardiovascular risks) would contra-indicate the surgery for anything but cancer or severe endometriosis/adenomyosis. Even my surgery at age 46 was considered "young" (I had a six-inch ovarian mass that was glued to all my internal organs and twisting them, markers and family history for ovarian cancer, severe adenomyosis that was causing profound anemia, and stage IV endometriosis). It was NOT a walk in the park, and I was otherwise quite healthy (a runner at the time) and not trying to recover from a complicated pregnancy.
If your husband is having surgery, and you have a tubal ligation, that's about as certain as it gets without a hysterectomy. Even tubals and the Essure procedure are not without their risks, but a lot less high-impact than having such a drastic surgery done.
If you do plan to do the tubal ligation during the c-section, make sure you take any required classes in time. I had to take a four-hour class at least one month (maybe even two) before the surgery; luckily I'd scheduled it in time since I did have a somewhat early delivery at 37 weeks. Most insurers (or maybe it's the state, I don't know) require some sort of "counseling" well in advance of the surgery. And, if you're pregnant now, your DH could have a vasectomy at any time (after his class, of course) during your pregnancy. Just a thought; that way, he'd be "ready to go" by the time you're ready to resume relations after your baby comes!
Good luck. I do know how tempting it is to ask for that. It never would have occurred to me, to be honest, and we did have one moment about two years after DD was born where I thought I might be pregnant for a day or two (but I wasn't). Even with my history and symptoms, I never would have had the hysterectomy had it not been for the risk that my mass was ovarian cancer (thankfully, it wasn't). And yes, my hysterectomy turned out to be a good thing (for me) in the end, but the recovery process and menopause process was long and not particularly easy or without complications, some of which still affect me.