Rarely does a patient get everything for nothing. That is, there are imperfections of the implant procedures that you will need to accept and reconcile this with want will make you happy. Since I do not have the advantage of examining you, I can only interpolate from what you have said so far.
It sounds as if your breast soft tissue have been stretched either preo-op or from the large implants that removing them will produce a saggy appearance, hence the suggestion of a lift if you shifted the implants to a submuscular position.
The texturization of the implants, especially on top of the muscle in someone who might have somewhat thin tissues to begin with will be prone to rippling and wrinkling and possible visible folds even if the pocket is made larger.
The pockets will ideally be only as large as the implant because the purpose of the texturization is to allow the soft tissues to stick to the implant surface. The shelf-like appearance is probably due the high profile nature of the implants and the size relative to your native breast.
If you want to get rid of visible folds, you should consider a smooth implant and/or submuscular placement. if you have a saline implant, consider switching to a silicone. If you like the size and shape of your breasts, then you might not want to go submuscular.
I am not sure what the motivation was to use textured implants. Perhaps it was the fear of getting a capsular contracture in the subglandular position with a silicone implant. I usually start with the smallest, lowest profile, round, smooth implant that fits the anatomical dimensions of your native breast and if this doesn't get you what you want, you can vary the dimensions and location of the implant.
Most patients will opt for submuscular because it allows for less risk of capsular contracture and better mammograms and less long-term soft tissue atrophy. There is no best procedure, only choices determined by the priority of your desires and assumption of risks.