If you want to travel and make a career on your own, you dont need the university. What you dont want is to become an employee. What you want is to become an owner or investor, as in: Run your own business or have enough money to live of the interest.
Passive incomes are the best friend of long-term travellers. Because you spend much time travelling and living cheaply, a passive income, even if it is not much, will carry you a long way. Renting out property, getting royalties from art, books, articles, music or games are a way, as is affiliated marketing, sponsoring and running a website with ads.
Working on the move is way more difficult. First you have to get a working permit, which is easy for Australia, NewZealand, Japan and Canada, but hard for other places. Second you need to work, which is no fun. Third thing is that you have to get to know the laws and taxes of each place you work in. OR you work cash-in-hand illegally somewhere, which has its own pros and cons. OR you do what is called WWOOF-ing, which is working for food and lodging, which is legal, because you dont have an income. This allows you to stretch the money you have further and increase your travelling time, but of course does not generate an income in itself. Good examples for this are a PADI Divemaster, you would live on a beach in a tropical region, get free housing and food and go diving every day for free. Sounds good, which is the reason a lot of people do this. (including my girlfriend who is a divemaster, and I am a certified rescue diver. Speaking about me: I also have a passive income from rent, sponsors for free gear, a website, the MDF donations (yeah, thanks everyone) and interest on stuff on banks. Everything just little drops that taken together are good enough to travel for 7 years now without having to work.
I dont think that you have to worry about a career in your age. I know people that have started their own business and went from completely broke to well-earning upper-class citizen in 2-5 years, all without any degrees from any schools or universtities. They are only needed if you want to work FOR someone. No one asks you for your credentials if you start your own thing.
As far as I understand it, being a student in the US would leave you with dept, which is kinda the negative version of that passive income I was rambling on about. Definetily nothing you would want if you really want to leave a life of travelling.
Suggestion: Since you say yourself that you "dont want to waste 4-6 years on a career path that is going to make me miserable", but say that you really want to travel, travel first. Pay no heed to nay-sayers and people that tell you its a bad idea. Because even if it turns out to be one, you will not be worse of for it. The worst that can happen is that you have to move back to the US, get a shitty job to earn money to study. Which, as I gathered, is exactly your situation now?
Just a personal note: I have seen countless career and job opportunities on my tours, have been offered jobs, and very well paying jobs as well... and I wouldnt even blink if someone tells me that I need to move to another country or learn a different language for it, because I do that anyway. It might broaden your horizon to know that every place in the world is nearby, easily reachable and could be a way better home than the home you are used to.
For example I would pay over 1000€ each winter just for heating in Germany. For that money I can fly to a warm country, stay 3 months on a sunny beach, and fly back to Germany. And I think it is a way better value for money.
Another good job is IT or anything you can do online. Be it blogging or coding, anything you can do on the fly with only a laptop and an internetconnection is a real winner when you are travelling. Again, I personally know many people that do that.
Question from me to you: Why Europe? Why the most developed, white-western, expensive, safe, easy, populated area in the world? (which only hands out 90day visas at a time, so you have to leave the Schengen-zone every three months and re-enter)