Now we know where the leak is. I don't think you need an new fixture. All you need is a washer. Readily available at any hardware store although you will have to buy a variety of them due to that's how they sell them.
First you will have to remove the faucet. Most likely on top the faucet is a plastic cap that says H or C. Remove that plastic cap with a small straight blade screw driver. Below that cap you will find a screw, most likely a Phillips. Remove that screw then lift that faucet off the fixture. It may be stuck and difficult to remove just keep tugging.
When you have the faucet in hand, at the bottom you will find another screw again most likely a Philips. Remove that screw and the washer it is holding in place. From the box of washers you got at the hardware store, select one that is the same size and shape as the one you removed from the faucet. Place it on the faucet and put it back together reversing the process described above. It's also advisable that you do the same with the other faucet.
BE SURE THERE IS NO PRESSURE IN THE SYSTEM BEFORE DOING THIS OR YOU MAY HAVE WATER EVERYWHERE!