FIRSTLY and most importantly, Liza, let me welcome you to our cozy l'il group of Glass Egg Lovers
As with ALL good paintjobs the best results ALWAYS start with the best prerperation job. I think I'd try to get as much of the old paint
off as you can, then make sure all areas you want to paint
have been 'scuffed', usually just a light
sanding, just enough to remove any sheen that might be showing. Following that wipe all surfaces down with a good cleanser (mainly to remove any possible fingermarks which might leave a slight 'film' to which the paint may not stick as well to. You might even consider a coat of primer paint (specially usefull to ease the transition between color changes from the old to new paint.
If you are still unsure what to do, why not come out to your local Glass Egg Meet on the May long weekend at Bigfoot
Campgroung in Harrison?? Many of us there have painted inside and outside our 'egg shells' LOL!!!!