I know the Lat Pulldown is a great exercise for the back, but whenever I do it, my shoulder starts hurting. Is there an alternative to it in without overhead movement, where the hand does not move above the shoulders?
I have the same issue. Rows, will work the lats very well. There are about a million variations on rows too, so try them and see what works best for you.
Quite frankly, I think the inverted row is one of the best back exercises one can do. It's body weight, but beginners can bring their legs closer to their body while more advanced folks can leave their legs extended. Once that's no longer challenging you can switch to a one-armed row too. Oh, and you need almost zero equipment. Maybe a broomstick and two chairs?
That said, look around here and see if there are any other back exercises you can do that won't hurt your shoulder. But, give the inverted row a shot. You might be surprised that it's such a challenging exercise.
Do some shoulder mobility work before starting your workouts. Take a pvc pipe or a long stick of sort, hold it in front of you and put it behind your back. Do band pull aparts. Do some rotator cuff work. Warm up.
You can do assisted pull up machine, yates rows (look it up), dumbbell pulls, high row machine etc. Do some dead hangs.
But I think your emphasis should be on checking out your shoulders, doing mobility work (look up kelley starr's becoming a supple leapord) and warming up.