The way I do it is to use a paint brush and a spray bottle. I oil both sides of the leather and get in as far as I can with the brush and spray. I'll pull the leathers down so that I can get at the part that wraps around the tree and I'll take the stirrups off to get at the leathers there as well. Be sure to get the bottom side of the jockey.
The only time I would not do the seat would be if it was suede, otherwise "yes".
Since it's been a while since you've oiled your saddle I would let it soak in over night. BTW, before I oil my saddles, I clean them with Murphy's Oil Soap first.
Try not to put on a heavy coat of oil, it's better to do several light coats. Let it dry for 12 hours or more if you can, then apply another light coat. If it's really that dry, I'd do another clean and oil session in a week, too.