Yes, cocobolo is oily, and a tung oil finish will remain tacky for a long time after application. The last cocobolo I did, I used teak oil instead and was happy with the results. I wiped it down with lacquer thinner immediately before application, flooded the surface and kept it wet until it appeared to stop drinking in the finish, then thoroughly wiped off the excess. After that, I kept an eye on it for about a half hour and re-wiped it as needed because some of the finish tends to weep back out, especially on end grain.
I also learned the hard way, that cocobolo, again due to the oily nature, can't be stabilized,,, at least not with cactus juice. I confirmed this with Curtis at Turn Tex.
Since that project, I've picked up a bottle of Tru Oil and was happy with the first use, happened to be on paduk. I should give it a try on a scrap of cocobolo and see how well it works.
Good luck