I wouldn't worry about dissolving the old LLC. You may as well keep it, especially if you plan on doing business in that state ever again. However, if you're not going to continue business there ever again, then you can just let it expire. That's how it works in my state, so you may want to double check with your Dep of Corporations etc.
In the new state, you'll need to re-register all over again. It will be as if your LLC never existed, because technically and realistically, it never did in the new state. Expect to pay for the registration fee and the name reservation. Other than that, you can keep the same Articles of Incorporation, with the alterations made for the new state.
Here, such a thing would cost about $100. Fees vary from state to state, so yours may be higher.
Thanks for the question!