As someone in the hvac field for many years, I can strongly suggest NOT getting a liner with oil, I've seen many stainless liners fail. They have to be taken out in pieces and chunks, sometimes impossible to remove thus ruining the ability to use the existing chimney.
Understand a chimney is under negative pressure along its length and a few bits of missing mortar between the Tile liner stack is not a safety hazard in an oil flue. It can be a safety hazard in a wood stove flue due to large amounts of flammable creosote that may be deposited; these deposits can seep past the liner and into the chimney structure where, during a chimney fire they can ignite raising the risk of a structure fire. This is the original purpose of a stainless liner and one of the few where they are useful.
It is rare that flue gas condensation alone is the cause of moisture on the exterior of the chimney; It is even more rare that it would occur in an oil burning flue. As has been mentioned, oil exhaust is hotter and drier than NG/propane.
If I had a dollar or two for every chimney I see where a roofer or a chimney professional could not properly flash it I could probably retire by now. Competence is a rare thing.
And in your particular case I can almost assure you that the cracks in the crown and the mortar cracking around the top of the chimney is the most likely culprit for your issue. The reason this almost always shows up more during cold weather has little to do with the flue exhaust and significantly more to do with the fact that the moisture that gets into the chimney from the crown no longer is removed through evaporation into the outside air. This moisture builds and builds and finally starts seeping out of cracks further down in the chimney structure leading to the issue in your attic.
Checking the flashing thoroughly and if there's any doubt hiring a competent individual to flash it properly as well as
thoroughly sealing the chimney with an appropriate polyurethane based caulking and then applying a crown and masonry sealer (there are products specifically for this on the market) to the top and sides of the structure, once complete, would probably give you many years without leaks.
The alternative and the almost permanent way to solve the problem with your chimney in its current condition, although you won't like to hear it, is to tear it down to the roof line and have a competent Mason or bricklayer (I recommend not a "chimney professional") rebuild it from the roof up with new flashing and mortar and a proper poured concrete crown. Please note that finding a good Mason for the small job may not cost much if any more then that liner which will do little for you if anything.