Is there anything I can do to stop my "high beam" nipples from showing all the time?

Thank you for your question.  It seems that you would benefit most from a nipple reduction surgery.  Please see your operating surgeon to discuss this possibility.All the best,Dr. ResultsMiami, FL

Based on the description, you may benefit from nipple reduction surgery. Nipple reduction is an excellent procedure for the patient suffering from prominent nipples. This procedure can reduce the height and width of the nipple, usually preserving sensation. It can be done under local anesthesia or in combination with other procedures (under general anesthesia).
I like to use the “sleeve” technique to reduce the height of the nipple. This procedure leaves the core of the nipple intact preserving nerves and lactiferous ducts. The suture line generally heals very nicely in this area. The width of the nipple can be reduced by removing a triangular segment from the tip area.
I hope this, and the attached link, helps.

HelloIf the problem is that the nipples protrude more than you would like, there are several techniques for nipple reduction that would be helpful.  Make sure you go to an experienced breast surgeon and that you discuss the possibility of decreased nipple sensation.  Best of luck

There are several methods to reduce prominent nipples. See a plastic surgeon with significant experience in breast surgery.