This might sound gross, but it's the best solution, and many athletes actually do this on a regular basis. Simply don't swallow your spit or spit it out, just let it go, continue running, and pay no attention to when you start drooling. Some athletes, especially runners, I know of have that problem where their saliva gets "hard" and tough to swallow and tough to spit, the only solution is just don't swallow since if it makes it hard to breathe it'll interfere not only with your running but may cause you to choke, and as you said, you're running a half marathon, so it will interfere with it. And it's the most natural solution (as weird as it might feel at first) since your mouth is naturally open while you're inhaling and exhaling rapidly to give yourself the air you need quickly to keep going. Nobody will notice it when you start drooling since you'll be running by so fast, or if they do notice, they won't care and may even be sympathetic because they may also be an athlete who gets thick saliva.
Some people have to drool when doing something athletic, to the football player, to the weightlifter, to the runner, you name it (but for some reason I notice this is more common in runners because the saliva more easily gets thick like yours). I hope you don't immediately dismiss my solution as "too gross", afterall, it could give you an intimidation advantage in your half marathon. Afterwards, once you catch your breath, wipe your mouth on your shirt quickly to take care of the drool.