You have to strip the surface of any sealer, waxes and oil. IF you can find anyplace like a real shoe repair or leather dye supply they have strippers for just such. I have used laquer thinner in the past. Start out with a good wash with mild soapy hotish water, let it dry a bit, then put on some rubber gloves and use the L.T. on a cloth to rub & lightly scrub. Repeat as needed, maybe over the course of several days as you want to be really sure you are bringing those oils up & out and getting them absorbed by the rest of the dry cloth.
Make Double Sure you dispose of those L.T. rags properly!!!
Let it sit and dry a day before re-dying. If you havent cleaned the pores out well enough the dye will go on spotty. That means return to the L.T. Be aware that once re-oiled the finish will be darker than the dye you choose, so choose color with care.
Once you get the dye top take and the color you want, let it dry in well and start to retreat the oils (I use PURE Neatsfoot Oil and 'seal' with a beeswax topcoat, buffed).
BTW, I saw an online video where the guy put his saddle in an oven on LOW to warm the leather to re-oil it. BAD, BAD, BAD; cookies go in and oven, and roasts, etc. NEVER leather.