Just delivered last month! My excess saliva issue finally cleared up around week 14. Chewing sugar-free mint chewing gums was my ultimate lifesaver during those weeks.
I felt so gross and frustrated with this spitting thing in my first trimester. For me, it started fading by week 16 and I literally cried with relief when it was finally gone.
Ugh, this ruined my initial food cravings so badly! It mostly gets better around week 13-14, after which I could finally enjoy my favorite street food without that weird taste.