Welcome to the forum!
The following is based on my experience with both: A direct vent pressurizes the flue and expels the products of combustion through the pipe at pressure higher than ambient. A power vent draws the flue gas out by creating an artificial draft, and also requires a barometric damper as part of the install.
I would recommend a power vent with the associated controls to perform a post purge to ensure all CO has been pulled out of the flue. I would also say that it isn't a good idea to have both vent into the same chimney even with the stack damper, and in fact violates NFPA regs and your local Authority having Jurisdiction (probably the fire marshal) will be able to answer for sure. Plus there is always what your insurance company has to say. Best thing is not to do it.
Power vents are usually available used on ebay or craigs list and would probably cost between $100 and $300 (with controls) for your needs.
Keep us posted on how you solve this situation.