So, the way I interpret the Skeksis is that yes, they have the same emotions that other creatures do, but how they process those emotions is a completly different story.
All of the Skeksis have been shown to express a wide variety of emotions. SkekGra abandons his conquering ways and becomes a Heretic with urGoh and is rather nice to Gelfling (even Hup, though there is still the whole “Podling Slave” thing…). He is caring and quiet when urVa is injured and isn’t afraid to voice his scariest of worries.
SkekSa is close with the Sifa Gelfling and (book spoilers) when she is betrayed by them, she takes it very seriously and is very, very hurt. Though she saw them more like pets, there was still a degree of respect that she had with them. (spoilers end)
SkekOk has a genuine fondness for Gelfling (even if superficial), comforted Brea when they arrived at the Tithing ceremony even though he honestly didn’t have to, and seems to have a good friendship with both skekLach and skekEkt (even being distressed and sad when the former died).
SkekTek expresses worry over the Darkening and its effects on Thra, is fond of animals (especially Sidetic), and, at one point, had an actual friendship with Aughra, who all the other Skeksis seem to really hate or, at best, tolerate.
And all of the Skeksis have their own cliques and friendships. Alliances withstanding, the Skeksis have been shown to have just… Rather regular conversations and playful jabs with one another.
Though they may feel guilt or concern, those emotions are often superseded by desire, fear, and greed.
My ultimate interpretation is that, while the Skeksis can feel regretful, the way they feel their emotions is skewed. They can be concerned, but a threat from the Emperor or a nice bribe will easily wash any concerns away, no matter how dire. They might feel guilty for murdering something, but if that murder works in their favor, what’s the harm?
And drinking essence made it so, so much worse.