What you are doing is good, and yes, the oil does make the hair stick a bit

just make sure you wipe the excess oil off with a rag, and before you use them, wipe the blade again so you don't leave any oil streaks on the dogs

I would definitely get the blade wash too.
If you have a clipper that takes A5 blades, you can move the bottom part of the blade from side to side (not all the way out tho, they are a

to get back in!) and use an old toothbrush to brush out most of the hair. Then use blade wash, you dip just the cutting part of the blade in the wash, while the clipper is running, for just a couple moments, then pull it out, blade still pointing down, you don't want to get the wash into the motor part of the clipper, and wipe the blade with a rag. Then you can oil.
Often when you have the blade in the wash you can hear the motor speed up, that means you've got it clean.
If it is not an A5 type but a screw on type, then you can use a pipe cleaner or a wire tie to push the hair out from underneath the blade, as well as using the brush like you've been doing, and then do the blade wash as above.
Use your brush, or an old toothbrush, or a blast from a hairdryer to get the hair out of the clipper itself.
Also don't wrap the cord around the body of the clipper itself, it eventually weakens the wires inside where the cord come out of the clipper body, then you have to replace the cord

just loosely coil the cord and maybe put it in a box.
Here is a good website with clipper and blade info, and if your computer is better than mine, some videos too. Just clip on "clipper blade articles" or "instructional videos" at the top:
http://www.thebladeguyz.com/home.html