The Logic Board "is" the phone, much the same way the Engine is the car. Everything else is just the Frame, Windshield, Battery, etc... So yes, you can change the logic board...but to remove it out of a working assembly would be a waste of time if you take a working board out of a working phone.
Some water damaged phones can be repaired too if done properly but the cost of repair for a 5S (Depending on the value of the Data) just doesn't seem worth the trouble either.
If you need further help, feel free to repost :)
Best of luck!