You are completely describing my son at that age. Today, at just-turned-5, he has virtually no separation issues at all. He's still somewhat shy/reserved when he first meets someone, but he has wonderful friendships with kids his own age. In terms of independent play, there are definitely times when he demands my complete attention/involvement, but he can also play or read on his own for about an hour at a time -- less than some kids his age, but well within the range of normal. And, unbelievably, he's finally learning to swim, and he loves it. He just had a semi-sleepover with his two best friends, while we parents all went out for dinner/drinks. He's not a perfect child -- far from it -- he can definitely be oversensitive, and maddeningly goofy when he's trying to avoid something. But he's sweet, thoughtful, bright/gifted, and more to the point, he's FINE.
Along the way, I think, I did a whole lot of things wrong and a few things right. Brief summary below.
What did I do wrong? Doubt my son. Worry endlessly about disorders he might have. Sign him up for a preschool that was too noisy/chaotic and not structured enough. Worry more. Obsess about disorders, etc. Stress out to no end. Worry. Freak out. (You get the picture?)
What did I do right? Switch my son to a Montessori preschool where things were very calm and structured (and which was way out of our price range, but that's another story -- we almost went bankrupt paying for this, and I swear it was worth it). *Listen* when the new preschool teacher told me that "normal" is very diverse at this age. Watch my son's diet and keep him off all dairy, which can exacerbate anxiety in young kids. Accept that my son is shy and cop a serious attitude when anyone suggests that shyness is a bad thing. And yet, seek out parents with quiet, thoughtful kids (and face up to my own shyness in doing so) and schedule playdates -- in other words, stage-manage friendships for a bit.
In short, your daughter sounds like she has a shy, sensitive temperament. She will probably grow up to be a shy, sensitive, thoughtful, introspective, insightful adult -- a wonderful person, in short. But she'll never be somebody she's not. It's totally normal (and if you're me, inevitable) to think in terms of worst-case scenarios, but I recommend accepting that your daughter's anxiety threshold is where it is right now, and trying to help her take baby steps. If she can handle daddy and grandparents, can you leave her with one of them for half an hour at a time? Can you fit one social thing into your week (say, an hour of Mommy & Me) so she stops associating the presence of other children with the prospect of your departure?
Okay, I'm babbling endlessly here. I clearly have an investment in this distinct (and wonderful, I swear) personality type. Send me a message if you have specific questions or concerns.