Is a vasectomy alone enough for birth control?

My husband is down to get a vasectomy, which is great. However, we disagree somewhat on whether or not we need additional protection - I say we do, he says we don't.

Even if he does everything "right": goes to a top urologist, waits the recommended time after surgery, gets his sperm checked when doc recommends... I still feel uneasy risking it.

This is due largely to the fact that a) we initially got pregnant by surprise when my copper IUD failed, and b) we live in Texas, which has become an extremely scary place for women's reproductive health. If there's even a percentage of a chance, I don't want to risk it.

This is made more complicated by the fact that I don't want to get on birth control again. Hormonal BC isn't an option for me, and I'm uninterested in another IUD. We're currently using condoms, which I'm well aware have a higher failure rate than condoms. That said, in the event of a condom breaking, we know immediately and have Plan B on hand at home. If an "oops" happens with just a vasectomy, by the time I've missed my period and realized it, it's already too late here in TX. Sigh.

Trying to determine if I'm being too paranoid in this situation, or if we do indeed need two BC methods.