When I was doing that, and they asked how much I was currently making, I'd add $5,000 to my current rate. When they said they couldn't beat that but would match it, I'd hem and haw, then say the opportunities looked better at that org and take it. Once they upped the ante $1,000.
Got fooled the first time (gave an honest answer), then wised up.
Good moves on your part.
If you are currently employed, you generally have some better leverage with negotiating salary for a new job.
I once took a job, when unemployed, which actually paid a bit less than my previous job, but I took it because of wanting to get back to work, and have the stability of a full time job. Plus I had a baby and wife at home at the time, and I needed a job.
I had been out of work for a few months, and getting that job, which was with a nationally known company, seemed a good move, even if it paid less than I would have liked.