My green cheek is very affectionate. She went through a "teething" stage in which she had to learn how hard was too hard when she bit, but now she almost never bites hard enough to hurt. She has never made me bleed or made a welt or dent, even when she was learning about how hard she could nibble on me without hurting me. I do not consider my green cheek louder than my daughter's parrotlet or louder than many cockatiels.
I have heard that crimson bellied conures are even sweeter than green cheeks. I have never held a crimson bellied, but I would like to some day. I have had personal interaction with a rosifrons conure. Rosifrons and painted conures are also supposed to be very sweet, sweeter than green cheeks. All of those I have held have been very, very sweet. The black capped conures I have held have been very similar in personality to green cheeks. These are all various pyhurrah conures, so they make similar vocalizations. (I hope I spelled that correctly. I always have trouble with that.)
I have a bourkes and a linnie and a green cheek. I have to agree that, while I love the bourkes and the linnie, if you really want a playful and cuddly bird, more green cheeks will fit that bill.
Is there a speciality bird store close enough to you where you could put a deposit on an unweaned baby and then play with it while it weans? That is a wonderful opportunity.
From your description of what you are looking for, I have to recommend the small conures as my first recommendation, although you may find an individual linnie, bourkes, parrotlet, etc. that would be perfect for you. The issue with parrotlets seems to be that if they are not played with nearly every day from a very young age, simply having been handfed does not guarantee that they will enjoy human interaction. However, I also have to caution that I have noticed the same tendancy in the small conures.
Poicephalus could also fit what you are looking for, but they are larger and their beaks are more powerful, and they tend to have many more fears. I wouldn't rule them out, if you happen to find the right one, but, again, that would mean spending a lot of time with the bird. Commonly available small poicephalus include brown heads, meyers, and senegals. In general, both the brown heads and the meyers are noted to be a little more gentle than senegals, but the individual bird is really what matters. I find poicephalus noises to be a bit higher pitched than small conure noises, so listen to the vocalizations because one may be more tolerable to you than the other.
Pionus are another group of species to consider, but I find them considerably louder than either pyhurras or poicephalus. My daughter's maximillian pionus hen is one of the gentlest birds of her size I have ever met. Pionus can love a snuggle, and they are good shoulder birds because they are gentle even though they are really pretty large. However, I find both my senegal and my green cheek love to be cuddled more than the pionus. The senegal and green cheek have always been pets. The maximillian was once a breeder, so perhaps that is not a fair comparison. Nevertheless, I have handled many pionus of various species, and none have "demanded" petting and attention the way my senegal and green cheek do. It may simply be that they are not as demanding a bird in general. Perhaps they enjoy it as much, but don't ask for it.
Lories are very affectionate. They have needs that are different from the seed eaters you are used to. Most also have a louder and more piercing vocalization, but I have heard that the goldies lorikeet is much quieter than most lorikeets. I cannot speak about the vocalizations of the following two lories, but the ones I have handled have been wonderfully tame: dusky and all species of Chalcopsitta.
My first recommendation would still have to be a pyhurrah conure.