My Alfredo sauce always ends up gooey
Please do see the excellent answer to the Cacio e Pepe question, which applies to your future efforts.

The recipe you link to is just bad, which is why it's not working for you.  Boiling the butter and cheese in the pasta water is wrong, and will always result in clumps of stringy cheese.  So the answer is to not use that recipe.

The tradition for all emulsified cheese sauce pasta involves tossing the hot pasta in a bowl with the cheese and a little pasta water.  Fettucine Alfredo is no exception to this; it just uses butter as well as the cheese.  In all emulsified cheese recipes, it's critically important that the cheese not get above 180F/80C, or you will get stringy clumps like you did.

Some examples of recipes that do work, and show this principle:


Serious Eats traditional recipe and commentary
Saveur
Memorie Di Angelina


Then go and read the other answer linked above, because all of those tips for cacio e pepe also apply here.