First off, these are all my opinions, I am only semi-qualified to give them, and they are not a recipe for success.
1. You're dancing to have fun, right?
For me, the end goal of dancing is to enjoy myself. Challenging myself to be better, to learn interesting variations, to syncopate my steps, and to connect with a lead/follow are all ways to vary my dance and make it more comfortable. I think that this will make it more fun. I don't want to get better so I can be the best. I want to get better so I can have more fun. All of the following tips are methods to reach this goal. If you're sacrificing enjoyment to get better you're missing the point.
2. Take a lot of lessons.
Take the lessons offered in your scene. Ask questions during the lessons. Don't be embarrassed by your questions. I can almost guarantee that they've been asked before and that someone else is wondering the same thing currently. Retake the lessons until you feel comfortable with all/most of the material. Take notes on what you've learned and review on your own later. Once you feel comfortable with the material and have become proficient as either a lead or a follow take them as the other role. I will go into detail on this later.
3. Social dance as much as possible.
Go to your scene's social dances and dance with people of all levels. Don't feel embarrassed to ask people who are more experienced than you are to dance. Don't say no to people, or avoid dancing with people who are not as experienced as you. Try to enjoy each dance no matter what. You're spending the next three-ish minutes with this person so be present and let them know you're dancing with them! That's why making at least a small amount of eye contact during a dance is good. Congratulating each other when something cool happens is also encouraging! Lastly, don't apologize, unless you've physically hurt your partner. Most cool moves happened from mistakes so be confident when you make a mistake. You might've just invented the next big thing.
4. TRAVEL.
I know that some people don't have the time and/or budget to allow for this, but if you have the opportunity and the means to travel, you should. Dancing with people from other places will help you grow as a dancer. You don't have to go to an exchange or a workshop (though they are super fun and I highly encourage it); you can just go to a weekly dance. Each dancer dances a little different and each scene dances a little (or a lot) different. Learning how people from other places dance will force you to lead and/or follow to accommodate these different styles. You don't have to adopt them, but you might want to consider how they change your dance.
5. Try to switch roles.
If you are pretty comfortable as a lead or a follow try doing the other role. It will break your brain briefly, but it will make you a more informed dancer. Knowing what's happening on both sides is fantastic and will help you understand what your partner is going through when you dance with them. You might even find you like the other role better. There's no way to know unless you try.
6. Solo dance!
I was at Atlanta Varsity Showdown this past weekend and one of the instructors said something along the lines of "You can't do with a partner what you can't do on your own." Which made us all giggle in fits, but on a dance related note it's very appropriate. Get to know how your body moves alone. Being self aware when you dance will allow you to connect with your partner at a higher level.
7. Actively and passively listen to music you want to dance to.
I know there is a lot of debate on what kinds of music swing dancing should be done to. I suggest trying to dance to a wide variety of things before deciding what you like. Then, listen to what you want to be able to dance to in a passive and an active manner. Listening passively (listening without wondering what's happening in the song) will familiarize you with the songs and the genre. Listening actively (thinking to yourself what the drums are doing, what the base is doing, etc.) will improve your musicality.
8. Know when you've hit a wall and decide how you want to address it.
I've hit a lot of walls. I'm improving and improving and improving and BLAM I plateau. Sometimes I even back track a little. It's really frustrating when you love dancing and are trying to improve. There are two ways I have personally addressed this with success, if you can think of more, go for it. The first thing I've done is danced through the walls. It takes a lot of force because you feel like you're going nowhere, but you have to trust that eventually you will get through it. The second thing I've done is taken a break. I'm not talking about a break from dancing entirely. I'm talking about taking a break from situations that pressure you to improve. I took a break from lessons and traveling because both of those things make me feel like I have to improve. When I returned to these things I was able to absorb more and begin improving again.
9. Try teaching.
If you want to teach and you feel competent, ask if you can teach a beginner class in your local scene. I didn't know what I knew until I had a room of fresh-faced baby swing dancers looking at me for direction. OMIGOD it is scary the first time. They are dance putty in your not-all-knowing hands. But act like you have a plan (better yet, have an actual plan) and make sure they have a good time. By then end of the first lesson you will have learned just as much about your dancing as the class did.
10. Remember number 1.
It's so important I'm bringing it up again. You're dancing to have fun. Remember to think about that during all the other steps. If you aren't having fun why the heck are you devoting so much time to it? Go enjoy yourself.