Forget about teaching English, just about any dummy can do that and earn peanuts for doing it.Use your Engineering degree and earn some real money, pay off your debts and see the world! And dont forget you are required to sign a contract to work a stipulated period teaching English which is probably the same in the Engineering field except the $s are in the latter.
Kereru in post 6 is spot on when suggesting you look into a mining job in Australia The mining companies such as BHP and Rio Tinto are desperate for skilled labour and employ people outside Australia to fill the vacancies. Generally you work in remote locations and live in mining camps. These are usually fly in fly out to your principal town/city of residence. Many work 14 days on and 14 days off. I have just returned from Bali and Thailand and there are numerous mine workers over there on the down time. And they live well as they are earning big $s. If this interests you use the net and track down some of the recruiting agencies who will give much more info than I can provide.

New Zealand is another option. With your degree, you could look at the Christchurch rebuild, geo thermal exploration (as a power source) in the North Island, mines on the west coast etc.
Thanks for enlightening me.
You are not a teacher therefore you have not the ability to teach English correctly. Many consider it an easy way to make money and as an extra bonus on a CV but fail to think of the kids they "teach". Did you ever suffer from a bad teacher? You are more highly qualified than most, add some practical experience to that and you will benefit tremendously in the future.
You want to spread your wings after so long in school, I appreciate that. However, the world will wait and the years ahead are for cementing the foundations on which to build your life. Take a month in SE Asia (about the cheapest place to have fun) then get back to reality.
IMHO
Dave
I am hesitant to apply for normal full-time positions without any experience of what life is like outside of SoCal.
As somebody who has lived in 5 different parts of the U.S., with the fifth being a seasonal tourist destination, I can vouch that living some place for a year as a "normal" (i.e., employed) person certainly gives you a better feel for that place than vacationing there. However, by that reasoning, you would come back from overseas travel still ill-equipped to decide where to live and work in the U.S. So, I'll add Teach For America to my previous suggestion. They are particularly interested in STEM education, and they cover a large range of sites across the U.S.
I agree with others who said you'll earn more with your engineering degree than teaching English. Perhaps look into resource/mining sector jobs in Australia - they are looking for people at the moment. The work hours are long, often in remote areas, but the pay is good and you'll get good professional experience that can lead to other things, you'll get good money, and be able to hack into those debts. And, the Australian culture isn't too different to the US culture, so it might be a good first step in terms of travel. (I'm Australian and have lived in the US for a couple of years).
I'm a Texan who worked and traveled for 2.4 wonderful, laughing, life educating years in Australia. Go there. Not just the east coast party scene.
You'll always be greatful as I am. And still laughing at the memories.
That was 45 years ago.
I've learned things through very slow travels, that aren't taught in school. Good warm folksy, other culture things.
Surely you can get an engineering job outside the U.S.
And pay your debt from there.
Be sure to take a book of paper checks with you. So you can send money to your albtross education loan.
You have to authorize bank transfers, in person, b4 you go.
That's if you have an account in your working/living country.
Travel outside the U.S. slowly enough to make friends and get the drift of other p.o.v. ways of life. Not just to "see" things that are so old you can't relate.
Most of us yanks think borders are the edge of outter space. Sad. *see profile
If you have a degree in environmental engineering, use that as a way to see the world. I'm an expat and there is an entire world of overseas jobs out there beyond teaching English. Some of the big engineering firms have branches all over the world; Aecom comes to mind, but there are plenty of others. There are non-governmental organizations (NGOs) that are involved in projects in your field, check out some of those. Check out the UN website, there are opportunities there as well. Have you considered a career with the US embassy? Talk to your professors and see if they can point you in the direction of some international opportunities.
I commend you for wanting to escape SoCal, not many people realize what a soul-sucking place it is. Pursue some international opportunities and get that career underway. Who knows, you might land a job abroad that covers some of your living expenses and allows you to tackle that debt.

With your experience and knowledge, what do you think I should do?
Catch a cheap bus to Mexico and do the WWOOFing thing.
Just get a well payed engineering job and save up money to travel?
If you have debt, you should pay it before you travel, but don't wait too long, very few people are able to pull off backpacking with a baby...
Teach English abroad? Any other temporary work recommendations or opportunities I should explore?
Yes but it's more expensive to go to Korea than to Mexico. Don't be afraid of exploring your own backyard. Don't go to Cancun, but explore your neighboring country.