You can get an almost guaranteed exotic engram by doing the Leviathan raid and opening the chests in the underbelly. You will get three keys by completing the Pleasure Gardens, the Royal Pools and the Gauntlet encounters. You don't have to complete the raid, just those encounters. I'm almost certain I've gotten one exotic per three chests every time I've done the raid. Once you have all three, you can enter the Underbelly and hunt down the chests. Most of the chests are guarded by four Watchers. They act like cameras, and when they see you or take damage they will start to call for reinforcements. The object is to destroy all four watchers before their call is completed. When done correctly, the door to the chest will open and you can unlock it with your key. Killing the watchers fast enough is not a trivial task. Find some friends to help you. Four is ideal, but it's not terrible to do with just two. A grenade launcher can two shot them, as can a rocket and other weapons. Grenade launcher is absolutely the fastest though, and the Prospector exotic grenade launcher is the best for that if you have it. If not, there are a lot of options. If you need help with this, raid an all, my clan can give you a hand. Feel free to add me: xaren#1489

The exotic engram from vendors do NOT count. They have to be from drops IN GAME>