Prices for plumbers, electricians, or any other tradesman will vary widely depending on area. That said, it seems to me you are struggling with a problem that is, to be blunt, taxing your abilities. In addition, you are messing with property that is not yours, and risking damage not only to your unit but to others as well. My advice to you is to give this to the landlord to decide on how to handle and stay out of it. And, for God's sake, in the future don't use caustic drain cleaners. As you have found, they don't work.
Changing one out varies by the brand. Some are quite easy, some more involved. Finding one that's an exact replacement may mean you don't have to change anything except the motor assembly. But, this requires both electrical, mechanical, and plumbing skills as the new one generally won't have a power cord on it, and you'll need to do that as well.
If it's one made by ISE, you don't have to change out the sink drain and the whole thing comes off fairly easily after loosening the screws, loosening the cam lock, and removing the circular split ring.
Drain cleaning chemicals often don't work and can be a major problem if you then need to take the pipes apart as they are caustic. Chemical burns or destroying tools can be an issue and a plumber may end up charging more because you did this.
I think that Drano® was made by a Plumber, Then when you use it You will call him to get your tit out of the wringer. He gets your money twice.
All I would use is hot water and plain Dish soap. If that don't work, the shop vac may clear the clog.
Drano® or anything like it , Does more harm than good. Just ask your plumber. He loves your money...