Depending on where you are located, it would benefit you to visit a quality coach. You can run a quick search on the internet to see who is close by (but being wary of the people and places like you experienced at your CrossFit facility), or you can view a list of gyms and coaches affiliated with Catalyst Athletics here: http://www.catalystathletics.com/cert/affiliates.php
With your exposure to power cleans and squats, the Starting Strength teaching method deviates from what is optimal for weightlifting. Even if you've experienced executing a normal squat as opposed to a low-bar back squat, the chances are that you've trained the low-bar -- more suited for powerlifting -- much more than the weightlifting squat during your Starting Strength training. This puts more emphasis on posterior development, whereas in weightlifting, we need a much more balanced approach to our lifts and development. There are also the flexibility limitations that tend to develop when executing powerlifting squats, which will make receiving lifts such as the snatch and clean and jerk -- and even executing simple overhead movements, not to mention the overhead squat -- a difficult task should you not have someone that can coach you through them.
To answer your question regarding which program to follow, I recommend following a program that will best suit your needs. Technique may be the most important thing to work on, but you also need to ensure that you maintain those strength levels so that when your technique begins to catch up, you're hitting the ground running. I know this isn't choosing one over the other, but every program should be for the individual based on the lifter's needs.
Therefore, if you're around a quality coach, I strongly encourage you to visit with him/her. If not, then feel free to post videos on this forum for some instant feedback. If you can't get to a coach and would like to post videos, I recommend you posting videos of you doing these couple of things so we all can gauge your abilities and make more solid recommendations: pressing (front and back), push press, squat (back, front, and overhead), power cleans (full cleans if you can as well), and what you know about snatches.
I hope this helps you!