I'm currently doing lat pulldowns and bent dumbbell rows as my lat exercises. But I feel like I need to do something to get at my lats better... my biceps are taking on too much of the work.
Maybe dumbbell pullovers? Are those more of a chest exercise?
I'm not sure what you would call the exercise I tried the last few times: it's like doing a triceps cable pulldown except keeping my elbow stiff so I'm using my back/chest more.
your biceps shouldn't be the limiting factor on back work, you need to get better at recruiting your lats.
Try and flex your lats at the bottom of every rep of pulldowns. Really flex those lats hard. Also pause for a couple of seconds in the contracted position.
A form cue that generally helps is to think about the movement as coming from your elbows. So instead of pulling with your hands, you're pulling with your elbows and everything above the elbow is just along for the ride.
The exercise you mentioned with the cables is called a Straight Arm Pulldown and it's a fantastic lat builder. Make sure you are REALLY squeezing the bottom of each rep for a couple of seconds. If you want to get really fancy, try doing three or four sets of straight arm pulldowns BEFORE your regular pulldowns. I guarantee you'll feel your lats working with that. Ouch!
hope that helps. Remember, the most important things are pulling with your elbows and squeezing the reps.