I think it sounds like she is comforting herself and coping well.
I think that if you thought you could drop her off and walk away without anything to show for it in her behavior than you were mistaken. But that doesn't mean it was a bad thing. Just different.
At this stage I think the baby did amazingly well. The more comfortable she feels and begins to learn you are coming back, the less you will see her carry the blanky, binky and suck her thumb. I think that to take her out quickly will just confuse her. Just make it a regular routine. Reassure her that you will be back after play time. I say give it a month.
I don't think that she will be ready at two anymore than now. But you can bet that your mom dropping you off with alot of strangers, in an unfamiliar building, with strange kids is stressful at any age. So, it is stressful, (un-needed? only you can be the judge), but just as much now as at two. Why would you think that it would not be? But lets be clear. You need a break and some time to do errands. She probably doesn't need the stress of socialization from other kids at this age. It is your need not her need. Socialization is highly over-rated and an excuse people use to put their kids places that they feel guilty putting them in the first place, but need some rationalization that they are doing it for the kids own good. Socialization? No, not buying it. You need a break, which is ok, but say what you mean.
My kids never attended day-care. And we have always homeschooled. Socialization has never been a problem for my kids with any age group of kids. My kids interact well with people of all ages from toddlers to grown-ups. You child can get plenty of socialization from just being with you and going to the grocery store.
It is ok to need a break. You are a SAHM with a toddler. You do need a break. I am glad that you recognize that you do. Just be careful where you put her and the amount of stress that you cause. It sounds like you both did well.
But don't be fooled that she needs socialization at this age. She doesn't. Toddlers seldom play with each other or learn to share at this early age anyways. So what was it you were trying to accomplish?
Socialization is learning how to act and interact with others. She will be learning what the other kids do...which includes biting, hitting, tantrums etc. as part of that "socialization" and when kids are older that "socialization" is called peer pressure. So remember that. There is nothing wrong with keeping her at home. You can teach her manners, courtesy and respect without all the socialization. If she goes to the park, takes a dance class or when she enters school she will get plenty of socialization. You and her will be overly socialized. You should cherish the time you have with her without alot of unnecessary interferences.
When I needed a break I send the kids to Granma's or send them out for the evening with Dad. Or get Granma to come to my house. I even hired high school girls to come to my house to babysit while I ran errands and had a break. You can do that also. College girls work wonders one day a week. And they helped out with some light house keeping. Then at least the kids were at home with a stranger. And I eased them into that by haveing the babysitter come and be with us in the house while I was there just playing with the kids at first. SO when I did leave the babysitter was a familiar person. And they were in their own house. Still getting lots of attention. And the girl loved it also. She got to play and color and do playdoh etc. So it worked well for everyone.
Just be aware of what price you pay for socialization she really doesn't need. You need a break more than she needs socialization. What about swapping with a mom you already know. One week at her house a day, the next all the kids at yours. I have done that with great success also.
Good luck with whatever you decide,
L.