The best way to find out if Lucy is eating enough is to weigh her. The average adult sunny is around 120-140g(the latter being a touch heavy lol)
She is still adjusting and you will have to work very hard to undo all the negative things she's become accustomed to.
Sounds like she was never taught step up or if she was, prefers your shoulder because that way, she has the control. If she is on a human hand she must trust they aren't going to try and put her anywhere she dislikes etc. On your shoulder she is able to ensure this doesn't happen. It will just require a lot of patience and work from your side to prove to her you only want her on your hand for fun and nice reasons!
With her eating, she was on a poor diet before which you have overhauled gradually since you've got her - I'm correct in this I think? (sorry Tia, bird brain!!!) That being the case, it will take a while for her to adjust but remember, all of us fill up the pellet bowl and the fresh bowl and find most of it on the floor and smeared everywhere!!!! The expression "eat like a bird" to me, is rather true. They DO stuff their faces and a trip to the electric scales usually backs this up, it's just that with the contents of what seems like the majority of the bowl on the floor, it's sometimes hard in the beginning to tell WHAT they've eaten unless you've been watching every second of the day lol.
As you've a trip to the AV coming up, they will be able to advise you and weigh her as well as checking she is tip top! Great you made the appointment because it will reassure you too!
Not playing with toys or coming out the cage are learned behaviours. Maybe her previous owner never took the time to introduce new toys and show her how they worked when she was little? Maybe poor Lucy got little or no out of cage time after the initial excitement in her old home and as a result, doesn't realise how wonderously fun it can be?
Consider all of this and remember she is a bit like a serious human toddler. Your job is to loosen her up and show her that her old routine or lack of, is a thing of the past!
Wishing you luck!