I'm almost 5 months on T (consistently). I changed climates at exactly the same time I started T, so I'm not sure if my drippy nose is a result of going from humid southern Indiana to drier western Montana or from starting T. I'm on my second box of tissues at the office, which feels weird, but hey, there are bigger issues in my life right now.
When I have post-nasal issues, I gargle with listerine a couple times a day to cut through the sludge. Imperfect, but it seems to help, plus nice to have fresh breath.