Painful thumb
General mountain dulcimer or music discussions
There might be two kinds of "soreness" we're talking about here. One is a function of the friction of the metal wire against your skin. The second is muscle soreness from having to push down on the string.
For the first kind of soreness, I would suggest first that you use new strings. New strings are softer and less like hard wire. Second, I would suggest turning your thumb sideways a bit to use the edge of your nail. Using your nail will allow you to slide up and down the string much like a noter and won't cause any pain.
For that second kind of soreness, lower the strings as @ken-hulme has suggested and rest assured that the muscles in your fingers will get stronger over time.
If you are playing mainly in a drone style, though, I would suggest using either a noter or your fingers. Either of those techniques will allow you to play faster and also avoid the soreness that seems to be a hurdle for you now. (I sometimes use my thumb like a noter in the way it appears you are trying to, but I am a chord player who sometimes just breaks into drone style for a verse or two. It is not my primary playing style.)