Have no idea what "wet with oil" is exactly to the OP but there should be a real film of oil present in the wake of saddle movement.
Good practice at shutdown is to park the saddle at the tailstock end of the ways, the least used portion of the ways on the vast majority of lathes. Wipe the ways clean with an oiled cloth before and after that move. 10EE way wipers are awful, you have to make up for the small deficiency present in a now seventy seven year old design by a brilliant man.
At startup repeat the oiled wipe cleaning step, dust has messed it up. Set the machine to a high feed rate and run it at a good spindle speed with the saddle stationary for a few minutes to make sure it is lubricated by the carriage oil system before movement. Oil is a lot cheaper than rebuilding a neglected 10 EE.