You sound like your mind is in the right place for sterilisation.
TBH, mumsnet seems very anti-sterilisation. And the UK in general, I've found.
There's this whole perception that if you find another partner, you're going to want to have kids w/him/her.
When of course, some people really don't. Some people feel that, if my husband and kids dropped off tomorrow, and I met someone else, I don't want to go there anymore.
I think 34 is old enough to know what you want and need for yourself out of life.
It's a VERY good idea to talk this over w/a professional, however, and perhaps even ask your GP for a referral to chat things over w/a counsellor.
I know a good many people who are sterilised, both men and women. Some who have had kids, others who never have.
Haven't met a one yet who regretted their decision.
My best friend was just sterilised last December. She is 39 and, although happily married, childless. Her husband is also childless, but she wanted to be sterilised b/c she never wanted it to happen, which he was fine with.
My other best pal is a single mum to a now adult daughter she had when she was 23. She never wanted anymore kids and had herself sterilised at 38.
I know several men who had vasectomies, my dad, BIL and FIL - all fathers of two biological children, as well as some men who wanted to remain childfree.
All were VERY sure they NEVER EVER wanted kids anymore when they went for their procedures. There wasn't a shadow of a doubt for them or any pressure.
Good luck in what you decide.