You likely have it right - the DWV (drain, waste and vent) pipes in a bathroom are commonly all tied into one branch leading to the "stack", but you run into all sorts of variations. So in your case, sounds like bathroom sink and kitchen sink are on one branch, tub/shower and toilet on another, which come together BELOW the blockage in the sinks branch. Blockage must be upstream of the connection in the sinks branch, otherwise you would be getting substantial backup and probably overflow in the lowest elevation drain when shower/toilet were used.
So - since evidently in the sinks branch, snaking with a plumbers snake would likely clear it out fine. Plumbers and Sewer and Drain contractors can both do this for typically about $100-150 range - you might also be able to do it yourself (if into that) by removing the trap in the kitchen sink (shorter distance to the blockage) and running the snake into the drain line going into the wall or floor from the trap. 10' snake would likely do it, 25' one certainly.
Other possibility is the two sinks have separate branches which then come together inside the house, then jointly run to the stack - so it is possible, in a certain configuration, that snaking will have to be done from the bathroom sink - but unlikely if the plumbing was done to standard paractice and code. Would take a pretty odd configuration of piping for a snake from the kitchen sink to not clear it.
The leak from the disposal may be because of the backup - if only leaking when that happens. They are not really made to sit with water in them, so commonly leak around the shaft seal or housing seals if that happens. Could also be - see where the drip is coming from - just be a loose connection in the piping at the garbage disposal.
If still leaks after the blockage is fixed, and not just a connection leak, then unless you are really into mechanical repair, cheaper and easier to replace the disposal. Absolutely the case if leaking through the outer casing because it has worn through the wear ring (about mid-height) from use. Commonly around $200+ to $400 range installed unless you get a really fancy one - but normal Badger Insinkerator or WasteKing for about $70-125 range will do fine for all but very heavy use (which is bad for the sewer system anyway), plus labor to get you to the $200-400 range, depending on area. Here is an Angies List article on the cosst issue for disposal replacement.
https://www.angieslist.com/articles/w...
