I know this is really, REALLY old but just in case somebody else ends up here looking for an answer I thought this questions deserved one. On the Surface Pro 3, as in other Surface models, the digitizer comes as an assembly with the screen. They must be replaced together. This is not to be confused with earlier models that also had a separate digitizer BOARD that connected to the main board, that would go out occasionally and also need replacement (Rarely, many of these were replaced for no reason simply because it was the cheaper part).
Now, about that assembly I mentioned. You must be sure to purchase the correct version. The early LCDs that were released were faulty and would not run properly on the default resolution. You are looking for a V.1.0 or V.1.1. 99% sure it is a V.1.1. Anything lower than that is going to be a problem. They are a different make and not compatible unless you have a very early release unit. Shop around and pay attention. Not all LCDs are created equal. The same can be said of touchscreen digitizers. Some of them you will plug in, go to test, power it up and it won't work. It will detect, load the numerous touch drivers(8 I think?) then nothing. Others seem like they have to "warm up" for some reason and will suddenly start working later.
The point of all this is to say you should buy an assembly from a reputable dealer and be sure that 1: the LCD is compatible and will display correctly and 2: the digitizer will function properly. I can not stress enough how important it is to get this right, because these assemblies are EXPENSIVE. At the time of writing this they are still around $300+. Thee is a reason you can find SP3s on ebay with cracked screens for under $100 all day. At that price, and because of the total unmitigated nightmare the SP3 is to open up and work on to begin with, and the fact that you may get all this right and still have issues because you don't have a way to properly calibrate the darn thing and have to send it into Microsoft to anyway which you could.. NAY SHOULD have just done to begin with!!
....sorry about all that I tend to get a bit worked up over this particular subject. It is just rediculous and anyone attempting this, I wish you the best of luck. Now get to work!