to lap, to suck
嘬 centers on taking in liquid or food with a sucking or lapping motion, like an animal drinking or a person slurping greedily.
嘬 combines 口 (mouth) with 最, which likely contributes the sound. The exact historical development is uncertain, but the structure suggests a mouth-related action.
The mouth 口 is beside 最, which looks like a person leaning over to drink. Imagine a mouth lapping up water from a bowl.
For サイ, picture a sigh of satisfaction after lapping up a drink: sigh -> サイ, and the mouth finishes sucking.