YABU. Just because you don't like the usage, does not make it invalid.
"coming out tonight?"
"sorry, babysitting"
completely conveys the fact that you cannot go out because you are looking after the baby and cannot go out because everyone else who could otherwise do it is not available.
You could say
"sorry, I'm looking after baby tonight"
But that doesn't include the information that there's no-one else available to do it, and in fact can imply the reverse, that when the DP is in and you could go out then you would not be looking after the baby.
You could say
"DP is going out"
And leave it to the questioner to infer that that means you'll be unable to come out, but it hasn't really answered the question.
You could say
"DP is going out, so I've got to look after the baby tonight"
But of course, that is exactly what the word babysitting means, so it's a lot of redundant words.