to taste, to sip, to spit, the sound of sipping, to surprise
啐 centers on a sudden oral action: tasting or sipping, but also spitting or a sharp sound of surprise. The meanings split between a controlled intake (taste, sip) and an abrupt expulsion or vocal reaction (spit, surprise).
啐 combines 口 (mouth) with 卒, which likely contributes a phonetic element. The exact historical development is uncertain, but the character is associated with oral actions and sounds.
A mouth 口 beside 卒: imagine a soldier (卒) suddenly tasting a drink, then spitting it out in surprise. The mouth action covers both sipping and spitting.
For サイ, picture a sigh of surprise: sigh -> サイ, as the mouth 口 lets out a sharp sound.