Your experience is par for the course I assure you! It can take some adjustment to not flair the elbows, but you do notice how no experienced players do that. It’s all about getting your thumb to swing under the second finger with a lot of flexibility. It’s one of the natural movements of the thumb. Mess around with it in slo mo and you’ll see what I mean. Also the third finger comes over the thumb and that is a lot easier of a motion that you may think. Again work with it slow motion being totally relaxed and soon you’ll see how easy it is to move your hand in this manner. Your elbow will be completely left out of the picture before long.
Sure we could do a lesson or two if you want. Hit me up on email.