Could the Universe really be 12.5-13 billion years old? My initial reaction is that "this must be wrong" and apparently that's a lot of people's initial reaction according to the article.

https://www.nbcnews.com/mach/science/universe-may-be-billion-years-younger-we-thought-scientists-are-ncna1005541?cid=sm_npd_nn_tw_ma 

My question is, is this study actually making scientists scratch their heads, and, 2nd part, if accurate, how could the 13.78 (or whatever it is) estimate have been wrong, as it seemed to be the result of more and more accurate estimates over time.

One of the two estimates has to be wrong.
Because some measurements are difficult to make with absolute precision, it often happens that there is some uncertainty about them. The half life of the neutron was once thought to be 12 minutes, but in the 1980s this was amended to 10.5 minutes. In the early days, C14 dating was often substantially out by comparison with dendrochronology and other methods. It depends on the amount of C14 in the atmosphere at any particular time, and when they looked into the matter, scientists found that this varies from one century to another. When they took that factor into account, C14 dates became much more accurate and reliable. Back in the 1970s the age of the universe was thought to be about 15 billion years, but that has been revised to 13.8, and as new data comes in it may need to be revised again. This is why gravitational collapse of the universe has not been entirely ruled out, despite the advent of dark energy, which is thought to be accelerating the expansion when it should be slowing down. It is possible that like some other measurements in science, the dark energy measurement may also need to be revised at a later date.