Re: Old Story-New Problem!
Take heart & don't get discouraged---I think you can solve this thing.
From your post, you indicate that the toilets are working fine; this indicates that the MAIN DRAIN is open, so the direct problem is that JUST ONE BRANCH of the drain system is blocked.
Try to think of the drain system of a house as a large tree with a large main trunk directing water into the ground; the other branches, both large and small also dump water into the main branch (trunk)--the main trunk rarely if ever clogs up; the smaller branches simply because of their smaller size are more likely to do so.
You said you had the plumber out before & he ran a metal snake thru the blocked drain; although it's expensive to call a plumber, this will no doubt have to be done again since the lye & Draino have failed to do the job.
Google "residential drain system" to get diagrams and info on how home drainage piping works---it's all by gravity flow (downhill), and especiall with the smaller diameter pipes, they are prone to blockage, even in the most modern homes.
If the house is 25 years old, there is more of a likelihood that metal piping (instead of plastic) was used for the connector drain from the sink to the main drain; metal drains made of cast iron or steel corrode & rust from the inside since they are always exposed to water.
Rust & corrosion of metal pipes reduces the inside diameter & thus is more likely to cause a blockage.
Is your drain system connected to a city (municipal) sewer system, or do you have your own septic system????
Do you or the plumber have access to the drain pipes (are the in the open where they can be seen in the cellar, or are they buried under a slab???
If you were able to watch what the plumber did last time, you may save some money if you buy or rent a metal "snake" to try & snake out the line yourself; there are usually "cleanouts" at various points along the sink drain that you can open for easier access.
But given the fact that you can't use the sink or dishwasher, or disposal, and unwashed dishes are backing up, the only option points to getting a plumber in there to snake it out.