Am I rushing to get into a PhD program, and should do a Masters first?
I am a PhD in the US and in my field Master's is completely useless, basically, a waste of time. You will still need to take classes in PhD, do TAship and many other things that are basically mandatory distractions and a Master's do not waive any of them.

I would recommend skipping Master's and go with a slightly less demanding university/group for a PhD. In a Master's degree you more often than not don't get a chance to do research. However, Master's also sets the bar higher, so simply doing any research in Master's won't substantially increase your chances of getting into PhD. You can still consider a Master's degree if you can find a research program, not a taught one.

In addition to the previous point, a high-ranked university does not increase your chances of being a successful PhD. PhD is more about your qualities and habits than the rank of your university. What a high-ranked university does, though, is imposing additional stress and competition, often putting you in a situation where you struggle to even be average.