Opening a new credit card will temporarily lower your credit score due to the hard inquiry associated with the credit application. Depending on what your existing score is it may be as little as 3 points or as much at 25 points.
Given the importance of a
good credit score for mortgage approval and competitive rates, I would suggest you don’t do anything until you buy your home. The interest you save with a good credit score is going to be much higher on your mortgage than the $4,000 on your credit cards. It is much safer to apply for a
0% APR balance transfer credit card after the big home purchase.