Thank you all for your great replies! Your feedback is a big help, and I so appreciate the time you spent to give me your thoughts! I will definitely give my propane torch a try, and also a hair dryer. Some of the replies lead me to believe that my underlying problem has more to do with air density. My stove is in a room that opens wide to another room... in total about 4,000 cubic feet of space -- kind of an open concept. Typical 8-foot ceilings, but about 12 ft. x 40 ft. I do not feel any reverse draft tonight, probably because of the warmer temperature outside. (It's 40 degrees here in northeast Massachusetts this evening with 7 mph wind.) Maybe installing doors between the two rooms may reduce the amount of space that the colder air would otherwise try to fill? A bigger concern is that I occasionally smell oil-burner exhaust coming in from the stove. (The oil burner uses a second flue in the same chimney.) Fortunately, the CO alarm has never indicated a problem. My furnace is 30 years old, so I suppose it's time to replace it with one that vents out the side of the house.
Might help if you insulate the liner in your exterior chimney. That would reduce the amount of heat you need to get the draft going in the right direction. Also, is there a difference in height between the top of the oil chimney and the top of the wood chimney? If not, might help to separate them vertically by 12-18” or place a metal plate vertically between the two... I did some research on condensation oil boilers with direct vent. When I was done, I decided that I’ll probably just replace mine with a conventional one when the time comes...
A properly built chimney will have multiple flues ending at different heights so to alleviate just such a problem as this...your stove chimney is downdraft sucking fumes from the oil flue...as was mentioned already, one of those flues needs to be extended, or separated further in some fashion.