Yeah I really don't know why they had her do that. But Duff Killigan did the same thing and he too was shown to be staying in Drakken's lair so I guess the calling in sick was a joke.
All I know is that based on the episodes I've seen it would appear that Shego lives with Drakken and for the most part goes where ever he goes.
In fact in the episode "Go Team Go" Drakken is even quoted as saying "Shego, as long as you live under my roof, you'll follow my rules. And rule number one: no secrets."
Geesh that sounded so parental, no wonder why Shego blasted him.