When people say that they are busy, (i.e. I'm sorry if I wasn't able to reply to your message, I was busy) I think it's usually BS. I mean, the only time you are allowed to say you are or was "busy" is when you try to hide something that you don't like the other person to know. Cases like surprise parties/gifts/favor or a situation that would make the other person worry about you if they knew what it was. (usually just in movies...)

Why can't one say, "I'm sorry if I wasn't able to return your messages because I was tied up at work." Other alternatives would be - "I had a vasectomy" or "I was in a coma and I cannot wiggle my big toe." Okay, those may be a little extreme. Try a little honesty...

"I was adjusting to my new work."
"I was a little excited and psyched in arranging my new place."
"I was reviewing and preparing for my big presentation."
Hell, even "I'm becoming serious with this new girl I met." is plausible.

Last rant would be, "I was preoccupied with (whatever) at the moment." and "I had a lot of things on my mind that time" borders asinine. THOSE will make you want to breathe fire on anyone within a 50-meter radius. (cringe)