I lean towards don't pay off the mortgage club, especially if paying off your current mortgage slips you out of being an all cash buyer on the next property.
Think of it this way, between your savings account income (1%) and your mortgage expense (4.25%) on a $100k loan, we are talking about a net expense of $3,250.00 or so.
In my market, to obtain a loan we often pay 1 point and $3,000 in expenses. A bridge loan will be fairly high interest (and closing costs), and a HELOC will be up there a bit too.
Plus, you MIGHT squeeze out a better purchase price with your all cash offer.
Overall, having the options of that cash handy seems better to me since your current mortgage is low interest.