Here are my thoughts having been racing for a few years....

You don't need carbon. Getting a really nice aluminum frame like a Cannondale CAAD9 will probably be stiffer than low-end carbon, and cheaper. I really enjoy the feel of a nice carbon frame, but that is out of your price range. You don't need full carbon. Just get something that fits.

Since you're saving some on the frame (FYI low end carbon race bikes are often in the $2000 mark), go for some decent components. It doesn't have to be Dura Ace. However, with Shimano groups, there is a good bit of functionality improvement when you move up in the hierarchy as well as a reduction in weight. So, for example, my Dura Ace shifters are a dream compared to my old 105. The 105 was fine for me to start racing on, but they were kind of a hassle. The throw was far and difficult. I had many more dropped chains. I hated the brakes, though they have improved the brakes since I got that group. I really don't have to think about my shifting anymore, and that is a relief. It is easier to keep in tune.

You might consider moving to SRAM if you're going to upgrade most of the group anyway (SRAM is still Shimano compatible with chains, cassettes/wheels). SRAM Rival is pretty good bang for buck. It's heavier than the other SRAM road groups, but it is functionally the same.

A CAAD9 4 or 5 will be a great race machine for under $2k, and it or something comparable in another brand will be an improvement over the Dolce, Allez, or a low end carbon bike.

Finally, getting a new bike will mostly make you feel a bit better, hopefully give you a good fit, but you still need to train the engine, as others have said. There are some improvements that can help you in the long run, as you develop as a racer. A better quality bike can help reduce fatigue by giving you a smoother ride. Aero wheels might help you go a little harder a little longer because you can save some watts holding a certain speed. Aero wheels might help you descend faster. A stiff bike and stiff wheels will also make accelerations feel crisper and you might notice that they feel nice on climbs, but a fancy bike isn't going to launch you from a top 10 finish to a win all by itself.

These are pretty subtle differences that mostly have to do with rider comfort. I surely don't put out enough watts to flex many frames or cranksets, but there is something smooth and a little less "mushy" feeling about a high quality bike. So, I might be very slightly more efficient due to my bike, but it is mostly just enjoyable to ride. There are plenty of people on heavier, less "fancy" bikes than mine that kick my butt on a regular basis.