Ok I'm sad to say the only place I've ever had carnitas is Chilis and I absolutely loved them. Don't judge me. Anyway care to share your recipe Houston? I've looked online but the ones from Chili's had like a lime flavor and none of the recipes I've seen call for lime, they all seem to call for oranges 
Someone shared a recipe on here for shredded pork that might be exactly what you're wanting. Would've been a while ago - at least a year I think. If the person who posted it reposts it, that would be exactly what you're looking to try I think. Great lime in it. I might be able to dig it up as well if I can figure out how to review my post history...
**edit: figured it out. It was a post for Barbacoa by Ali from early 2014. It may not be quite the pork experience you're looking for, but it's -really- tasty. Reposting AtbC's post below:
crock pot shredded beef (barbacoa)
Total Time:6 hr 30 min
Serves 8
Ingredients
1/3 cup apple cider vinegar
1/2 yellow onion
2 tablespoons fresh lime juice
5 garlic cloves
4 teaspoons ground cumin
2 teaspoons oregano
1 teaspoon ground allspice
1 teaspoon salt
1 teaspoon pepper
1/2 teaspoon cloves
3-4 chipotle chilis in adobo sauce (I used ¾ of a small can)
3/4 cup beef stock
3.5 lbs. beef chuck roast
Instructions
1
In a blender, combine vinegar, onion, lime juice, garlic, cumin, oregano, black pepper, salt, cloves, allspice and chilis with adobo sauce and blend until smooth.
2
Trim any excess fat off the roast and place it in a slow cooker.
3
Pour the broth over the meat, then pour the pureed sauce.
4
Cook on high heat for 6 hours. If you want to cook it longer, do it on low for 8 hours.
5
Shred the meat in the juices and serve hot.
Notes
If freezing, make sure you retain all the juices. I put my leftovers in large gallon ziploc bags.
