The hip is an amazing joint. It can move the leg back and forth (as in walking), it can move the leg toward the body or away from it (as in standing with your feet apart), and it can rotate the leg inward or outward (as in walking “pigeon toed” or doing ballet stances). One of the first things I learned about muscles in college was the saying “Use it or loss it”. If one does not do a movement/function for a while, you lose the ability to do it. It sounds like, you cannot do the leg movement of raising your leg out to the side and probably the muscles that do this (hip abductors) have taken a vacation. Also the ligaments can get tight if not kept stretched out with regular use. There are different muscles working in combinations for all the complexity of walking. It often happens that when it is really hard to walk especially with balance issues, people do the simplist way which is very simple to the front and back movement and tiny steps (shuffle) because this gives more balance. This is not efficient nor “like before”. These are the exercises that I have seen used for hip abduction. Lying on your side and raising your leg up as others have said. I do this in bed because it is hard to lie on the floor and then get up. This takes a good deal of effort to raise the leg up. A simpler way is to stand holding on to a chair (I prefer the kitchen cabinet–because it will not fall over!) and raising the leg out to the side–alternating sides or doing one side and then the other. Another I have seen is standing bent over like you are a baseball catcher with feet as far apart as you can and then shifting sideways over one leg then the other. Personally, I would still go plop to the ground on that one, so I do it standing (at first holding on) with feet apart about two feet and then shift weight to the left leg so that it is straight up and down and the right is angled out from the body and then shift to the right side. This helps strengthen the muscles that pull the leg in so that one does not fall if both legs are not under you. You could also try as an exercise to walk with your feet further apart–this will strength muscles and balance. I naturally now have a wide-based gait.
Dancing is good for hip movement and balance and if you and you honey like to dance, it is a good activity to do together–especially slow dancing!
I have narrow feet and it has helped me a lot to wear walking shoes with a flat wide botton so that my feet stay flat and balance is better.
I know that I would not have been able to have weights on at first. Try to gently get the hips to work better first and then maybe try the weights.
Finally, if this is a big stumbling block to get back to normal walking, ask about more physical therapy. It seems to me that the goal is to walk normally–the way you used to walk. Our bodies are really smart and naturally do things in as energy efficient way as possible–this included the wiggle in the walk.