Major congrats on the weight loss. Even better, it sounds like you did it with a good plan instead of lots of running, water, and lettuce like some people end up doing.
I think pictures would help give a better reference point when you can get some up. Could be a few things. What has your training looked like? It could be as simple as not having done enough back training or doing too much chest training up to this point. In any case, posture is more of a skill or habit, rather than "just" being muscle-related.
Of course weak postural muscles play a role, but you can have strong muscles and a poor postural habit, and still end up with crappy posture. My favorite and simplest analogy is to imagine you're always wearing a cape, and try to make it flow when you walk. Or, you could actually wear a cape.
You should be able to do facepulls with the cable crossover setup. I'd start your workouts with 2-3 sets of 8-10 reps, using a pretty light weight and holding the contracted position for a full 5-count.
I'd pull a little higher than that ^ , so your hands are ear or eye-level.
The Neanderthal No More program, like Yogi suggested, is solid and helped a lot of guys. But as a first step, I'd work on those facepulls. Or you could try a 1-arm elbow out dumbbell row (different from the standard dumbbell row for back). It's the same basic movement pattern: