Do you have a P-trap or another type of trap under the sink, like a "concession trailer" trap?
Does the water eventually slowly drain out or does it stay?
Your in-line valve is one of those one-way valves under the counter to let air into the pipe, but not out, common valves in houses and RVs...
Sounds like the issues is after the "in-line" valve. Even though your snake made it through you could still have a blockage that just filled back in after the snake was removed. Try snake again and really work it and spin it in every section.
The valve others are talking about is unfamiliar to me so that could be it too, I cant help there.
Try a garden hose with trigger nozzle, spray water down the drain and this might do more than the plunger did. Risky, but it might dislodge anything. keep a hand or a towel over top so nothing sprays back in your face. use short quick burst of spray at a time.