I am a 33 year-old male and I have the same problem whenever I run over 5 miles.  I use Transpore Medical Tape by 3M.  You can get it at any drugstore.  The tape does not hurt when you take it off like Band-Aids sometimes do and it is a cheap fix.  There are also NipGuards but I have never tried those.

You might also try a different shirt.  Some people swear by Under Armor.

I have an Under Armor shirt, and it never irritates me.  But I got a different brand shirt a few months ago and went on a longish run with it, and suffered from irritation/bleeding.

I prefer an Under Armour compression shirt as opposed to band aids or body glide. Just my preferance and the shirt helps wick away the sweat. I had the same prob and it hurts for sure. Good luck!!

Thanks Drewdin, Being 58 yrs old and just starting this journey a little over a year ago has been very rewarding so far and have some more goals to achieve yet!!

put liquid bandage on your nipples.  It forms a protective shell that will protect you from the chafing/bleeding.  Much better than band-aids because it won't come off with sweat.

Body Glide works great, also for other chafing on long runs on hot days.