What is the plan for your current house once you are in your new forever house? If you plan on selling, then IMO it makes no sense to keep paying the extra on your existing mortgage. That money could be better spent on saving towards your new down payment.
Another thought, even if you have 20% down, you will still need to show that your dept to income ratio is good with both mortgages for a lender to lend to you...otherwise your new purchase will still be contingent on you selling your first house to bring your ratio down. If you make bank and can cover both mortgages, then this isn't an issue for you.
Using a basic savings account to save up for a down payment is not doing you any favors. The best savings accounts pay around 1% interest. You could have the money invested in funds that have a much better rate of return and are still liquid enough to be used as a down payment when needed. Even making 4% return verses 1% ends up being significant on larger sums of money (over $300 difference on $10,000 invested for a year).