I would wash it well with Murphys Oil Soap, and that will remove a lot of the gunk and stuff, rinse off and then saddle soap well, then finish with a touch of Passier Lederbalsm, rubbed well in, and then buffed.
Good news though, if your saddle is grabbing and holding onto dirt, it is probably grabbing and holding onto you
I'd only add a little detail: After doing the cleaning, let the saddle dry, then run your hands over it. It should feel clean and smooth. There should be no tackiness, stickiness, or gunky feel. If it's not clean, clean it again. When you're satisfied that it's clean, then do the Lederbalsam. I like to let it sit on undisturbed for about 24 hours before buffing it out, especially if the leather is dry and needs extra conditioning. Apply with your hands, buff with a soft cotton cloth, not a brush. Then get a fresh soft cloth and keep buffing until it's nice and smooth. It may still feel a little tacky for the first ride or two, but it will be smooth and sleek very quickly.