hi and welcome,
infection is transmitted primarily via unprotected sexual intercourse, contaminated blood transfusions, hypodermic needles, and from mother to child. Bodily fluids, such as saliva and tears, do not transmit HIV.
as you have mentioned using contaminated tooth brush will not lead to cross infection but you should avoid such practices.
As for your oral symptoms you can visit a dentist
and get the examination done for gingival infection, thorough deep scaling and root planing will help in relieving sympotoms.
use antiseptic mouthwash chlorhexidine
for oral rinsing which will help in healing the sores.