The Boardwalk is a good option. I would second Jellyrolls there. I can be a lot of fun. You can also stop into Abracadabar either before or after for a really well made cocktail and fun decor. Actually you could start the night at Flying Fish which is one of the best restaurants on property. (I LOVE the art deco interior Atlantic Dance Hall, but its days of being a true nightlife destination are sadly dead).
If you enjoy Tiki bars as I do, Trader Sam's Grog Grotto feels like the most adult Disney location on property to me. Waits can get insane on the weekends and there are no reservations, just a waitlist. But if you can find a night to go during the week, its usually pretty manageable. If you do have to wait for a while, you can get the same drinks on the outdoor Tiki Terrace (just no signature tiki mugs) and listen to some live music most nights. Or go upstairs and have a drink at Tambu Lounge, including one that you drink out of a big pineapple. But the antics and effects inside of Trader Sam's make it fun and lively. Must-do for me.
I think your success rate at Disney Springs will vary. You could get drinks at a bar like Jock Lindsey's, House of Blues, or The Edison. But the Springs is also a family destination, so in many of these locations you may also see kids and families (and if not in the bar, they will def be crowding the walkways) so it might not have the typical "nightlife" feel you want. You could always splurge for a bowling lane at Splittsville, where you can order food and rink to your lane. Again though, there will likely be kids.
If you're looking to actually go dancing, you'll have to take a car to one of the clubs in downtown Orlando. If only Disney had kept Pleasure Island open!