Had the EXACT same problem!
Tub was SLOW! Bet soon your other items will slow, too.
Tried everything, including a couple of bottles of coke (which I had
work in another home once)
I finally gave up. Tried snaking, DIDN't help?! Uh, why not? Seems
the drain piping fills with a clear jello like substance that the
snake went through, then came out and the jello remained!
SOLUTION -
Buy those hose attachments for for pressure cleaning out a drain. They
expand inside a pipe as the water is turned on to automatically form a
seal. Small one for small vent pipe, large canvas bag type for big
vent drain pipe. Don't be surprised at the price of that large one!
Get a pair of walkie-talkies, station one person inside the bathroom,
the other goes up on the roof and feeds a garden hose down the vent
drain pipe to just below where the drain feeds into the vent pipe.Turn
on/turn off water and see if safe to proceed. If so, do again, if
still safe blast away!
When I did this, the first 'nudge' backed the sink up slightly,
luckily the second nudge cleared enough so that the full blast really
cleaned the pipes.
Before, the water in the tub ran out faster WITH the plug in, After,
the drain speeds were like a wet-n-dry vacuum.
Years later I had to hire pros to clean the jello out from the pipes
going from the house to city mains in the street - that's large
diameter pipe, too! They snaked twice with pro equipment from the
clean-out access beside the house to the street, less than 100 feet
away and NEVER CLEARED that jello like substance out. They came back
the next day and used something the size of a jet engine mounted on
its own trailer, and sounded like it too, to blow the jello into the
city mains! Took them half a day using that tool and charged $600!
But, it did work permanently, well at least for another 15 years I
know of.