It will be like prohibition; on the outside the conservatives will act happy but if the situation arises they will get abortions. Just like when their grandparents wanted a drink - they did so. A system based on states still providing abortions will grow up. Some die hards will try and interfere with that system and most police departments will 'not prioritize' anti-abortion' investigations.
I've been thinking a bit about this, and I think this is an important point. Much of conservative support comes from those who are convinced that "liberal" government is meddlesome, complex, and expensive. But that's really because repression is applied only to certain parties.
I think one way around this would be malicious compliance. If the state owns your uterus, demand in writing permission to use it. Demand written guarantees that your gynecologist may have access to government property. Demand that, if it's illegal to cross state lines for an abortion, the state issue a paper granting you permission to cross state lines without search and seizure. Demand a written declaration of what measures are required or forbidden for a fertile woman to perform on her body in anticipation that an embryo that does not exist but might at some future time will find a welcome. Everything should be in writing. If the state wants to own our bodies, then it damned well better put its money where its mouth is. That should include by default that any medical procedure involving the reproductive system that is required for the health of any person should be the sole responsibility of the government that has asserted ownership.
In a state like Texas, where every neighbor of every potentially pregnant woman apparently has standing to initiate a lawsuit based on the suspicion of abortion, every potentially pregnant woman who is not planning an abortion should be able to phone every neighbor every night to assure them of this fact.
If birth control is made illegal, men should demand in writing permission to wear socks to bed. Demand written codes on what it is and is not legal to wear in case of sex, and be prepared to file complaints against anyone suspected of not following the codes. Hat colors, pajama styles. There should be a written guarantee that premature ejaculation, for example, is not grounds for criminal charges. Demand written clarification of whether vasectomy is permitted. After all, vasectomy is as much a physical barrier to the passage of sperm as other birth control.
Demand that agencies be established to codify and issue permits for activities, and that specifics be established for all of them, and printed. The administration of these laws should be expensive and complex.
If stupidity carried a financial or inconvenience expense to its promoters even a fraction of the emotional, spiritual, cultural expense it imposes on its victims, it would be dead in the water.