When it takes DD's Mom a while to respond back to an email I generally understand that she's busy being a wonderful single Mom. Especially in the beginning the SW passed on info to me that she was busy and adjusting.
Keeping that in mind, busy is a fair reason, but used enough can start to feel like a brush off if used to often. I hope that doesn't come across as rude, I truly do understand that you are certainly busy right now. I just thought maybe you'd like to hear the perspective of how "busy" reads to a bmom
In response to one of the other possible suggestions, I don't think that making it sound like it's sooo busy and sooo much work will make placement any easier or a bparent more secure in the decision they made to place. But maybe that's just me and because I personally didn't place because I thought it would be hard work.