Welcome to the forums - I'm not sure how you got into a loan that has you owing $11k on a 2012 model year 100k mileage vehicle, but it sounds like you are paying dearly for that choice. You can maximize your return by selling it directly rather than trading it in, but with "issues" that's not going to be easy. Most dealers will play with the numbers (discounts, rebates, etc) to get an underwater vehicle rolled into a newer vehicle, but that will just delay the lack of equity and probably put you further behind the 8-ball.
The best advice I can give you, and I'm sure it's not what you want to hear, is to double up or triple up on your payments to get the loan balance down at least to a break-even trade in range. If you just roll the negative equity into a new loan - assuming you can find a lender, the payments are going to be higher and your interest rate is going to be high too - that's a disaster that'll haunt you.
You'll have to look for something with massive rebates right now. It won't so anything to get rid of the negative equity, it will only make a new car approval easier.
Another option if you can afford it, just go buy a second car. When you get even on this car sell it.
Sounds like you bought a used car on a 6 year loan. Never do that unless you have a really low interest rate where you make more investing the money than paying it off. Long term loans will cause your situation where you remain underwater on the loan for a large portion of the loan period and seem designed to keep you in that loan while paying all that extra interest.
You may want another car, but make sure you can afford a much shorter period on whatever you get. I personally prefer 3 to 4 year loans, only going longerwhen I get a really low interest rate like I mentioned above. Usually the temptation to get a longer loan is killed when I look at how much extra interest I'll pay over the entire loan period. If that is the only way I can afford that vehicle, than I start looking for a cheaper car as I'm clearly buying more than I can afford.
I would not rollover the loan when you still owe more than half what you paid for a vehicle. You'll be even more underwater on the new vehicle than you are on the existing one and if the same thing happens, you'll be in even worst shape to deal with it. I would recommend making larger payments to reduce what you owe on it by about half before trying to sell/trade it. Also, interest rates have been going up over the past year and I don't see an end to that. So unless you have better credit than you had before, you'll likely be paying more in interest on the new loan which would incline me to keep my existing one.
You also may want to look for a cheap 2nd car that is reliable and buy that until you get your existing one paid off as it gives you something else to drive when the existing one has issues. I know there are cheap reliable cars out there as last year I sold my last vehicle to my brother for $800 since that was what the dealer wanted to pay me for it. But it was a very reliable vehicle and my brother is totally happy with it.
I owe $17500 on a 2013 toyota avalon limited hybrid with 162k .....trade in is $7500
its all relative