When I bought my truck, this was the situation. The previous owner had a lien, we met at the bank, I paid the bank, the bank cleared the check and signed the title over to me, the seller took home the cash. If you do this,
1. make sure you are getting a cashier check, the seller's bank would not take personal check for this (and get it done in same day).
2. Go early. I went at like 4:30, ended up being their till like 6:30, the poor bankers where not happy with staying late. Make the buyer come early.
Talk to your bank, they have seen this many times before. Its pretty seamless.
Congrats on kiddo #3