I would go to the GYN and endocrinologist first, if for nothing else to get a baseline of where your hormones are at.
You can try to eradicate the HP but the Antibiotics are a bit harsh but some people are not bothered at all by them.
The best 1st line treatment is a triple therapy of clarithromyacin, amoxicillin and bismuth citrate.
Be very careful because some doctors will mistakingly want to add a PPI right away. The reason they add this is to lower acid way down so they can get past the mucosal barrier. It usually ends up backfiring and will cause damage to the stomach (as PPIs often do). Bismuth is harmless.
I know that you had some bad experiences with Antibiotics and PPIs before but if you think that your gut can handle those two with the bismuth you may want to try.
THe 1st line of this therapy cures about 75 % to 85% of those infected. If you are in the 15 to 20% who were not cured then you will have to go to a 2nd line of therapy. Only if you are in the 8th percentile that doesn't get cured with that line will you have to go to the 3rd line. This is where they will insist on adding PPIs. If you aren't cured with 1st or 2nd line then I would stop there.
It may be worth a try. Still please make sure that you get a check up with your GYN and get your hormones checked thoroughly.