This is an horrifically bad idea. The reason it's so hard to find a doctor who will remove the healthy uterus of a 20 year old is that it's such a bad idea. That means that most docs who would do such a surgery aren't very good ones.
Other reasons why this is bad: you think you're solving a problem by getting your uterus removed, but what you're doing is giving yourself some new, possibly worse, problems. For example, the cervix and uterus play a role in sexual pleasure and orgasm. Remove them, and you may be causing yourself to have problems reaching orgasm, or making your orgasms less intense. The uterus also is attached to a lot of internal muscles, meaning that after you get it removed you might have some problems with abdominal strength and with incontinence. Maybe you won't have those problems--no way to know until after surgery, and at that point it's too late, you can't go back.
If you were to get your ovaries removed, you'd sign yourself up for a shortened lifespan. Women who have had their ovaries removed before age 40 have an average of 8 years shorter lifespan, according to the Harvard Nurse's Study, and also experienced increased mortality and increased problems like osteoporosis and Parkinson's. Yes, you can take hormones to help to combat that, but you have a naive attitude about hormones--they're not as good as the real thing, and it can take months or years to get them as you'd like, if that is possible--I know young women who have gone through surgical menopause who never have been able to get their hormones right, which affects their quality of life, 24/7/365. If you leave your ovaries in but get your uterus removed, then some women go into menopause only shortly after the hysterectomy. Then yes, you'd have hot flashes, night sweats, and, here's the really bad thing they just don't tell you about but that has been well-documented: your VERBAL MEMORY is negatively affected. That is, estrogen, made by the ovaries, is very important as a neurotransmitter in your brain. It helps your language abilities. If your ovaries are affected by the surgery, then you will have problems with language--something that should terrify anyone wanting to be a writer. I say this from personal experience. I went through surgical menopause at a young age, and the immediate effect on my language skills was very obvious to me. They'll often say to women that it's just because they're getting older. But I was young, so such an explanation doesn't work. When I say problems with language, I don't mean you won't be able to speak or something like that. Rather, the sharpness of your verbal abilities may end up blunted. It's not something that would necessarily bother your average woman. But for a writer, you need to keep your rapier wit at its sharpest. You don't want to sit there puzzled, unable to think of a word. Before surgery, having a word on the tip of your tongue is an uncommon occurrence. Post surgery, it will be common.
Also, some who get a hysterectomy but keep ovaries may have a period for a few years then the ovaries conk out and they have menopause, while others will just go through menopause a few years early. I'd be fearful of the quality of the surgeon, who would remove a 20 year old's healthy uterus, so in that case you are up for more complications, potentially. Surgery is not a magic cure-all, which is why responsible docs look at it as a last resort rather than as a way to work around anxiety issues and deal with menstrual pain that could be dealt with in far less invasive ways, such as via continuous birth control to stop periods plus anti-anxiety meds, which are reversible and not dangerous, unlike major surgery. It's also fraud for a doc to claim something is necessary when it's not. So it might not be a good sign of their integrity that they'd be willing to do that.
When you have a hysterectomy, there's no reason to remove your cervix, really. If they leave the cervix in, then no, your vagina will not be shortened. They wouldn't do the thing you describe--shorten the vagina then reconnect the cervix. Cervical cancer is mostly caused by HPV, so if you've had the Gardasil shot, then that will for the most part protect you. But if your ovaries conk out as described above, then if you have a cervix, you'll need to take progesterone pills in addition to estrogen. If no cervix, then you only have to take estrogen--an important issue as exogenous progesterone potentially comes with a lot of nasty side effects.
Anyway, please think about what I've said here. Check out the website www.hystersisters.com, a website devoted to support and info for women who have had hysterectomies, because there are so many complications and problems that an entire website is needed to deal with it. You owe it to yourself and your health to try some other methods of dealing with your problems than surgery to remove body parts.