Congratulations on your engagement!
I hyphenated my name with marriage #1 and I really found it oddly disturbing. When I changed it back after my divorce I felt to much better. It was weird; I did not anticipate that at all.
So... that was one of the reasons I did not change my last name when I remarried. Other reasons: I have a very common first name, but a cool and uncommon last name. My whole professional identity is tied to this name. I wanted to retain a last name connection to my kids from my first marriage (they have the hyphenated name). And... my DH's first wife and I have the same first name, and SHE changed her name when they got married. That was just a little too weird for me! Even so, lots of people didn't even realize he'd gotten divorced, since we have the same name, so it's a bit bizarre.
I think DH was initially disappointed, but he understood all of my reasons (especially the ex wife one). His family doesn't entirely understand my reasons, they are pretty traditional. Well, they understand the ex wife one pretty well. The rest they don't get. They often call or refer to me as Mrs. ____ and that's not a problem. It's not like I'm not going to answer to it.
As for context, among my friends and acquaintances, some have changed their names and some have not. Most who have changed made their maiden name their middle name. Many of my kids' friends have parents with different last names; it's never been a problem in any of our schools. On the other hand, among my husband's friends, I am one of the only wives who has not changed her name.
I don't know if you and your child have the same last name now, but another factor to consider is how you will feel having a different last name than your child - or children, if you and your fiance want to have some of your own. For some people that's not an issue; for others, it's really important.
Whatever your decision, good luck!
ETA: Another option, which was sort of mentioned below, is to choose an entirely new last name that you will both/all share (assuming that your DD's father will allow her name to be changed? you alluded to changing your names together) - you, new husband, your daughter, and future children. I've had a couple of friends do that. It wasn't a combination of their names in any of those cases. One couple picked a name that they thought represented them well; they have since divorced but they kept their "married names". Another looked back through their family trees until they found a last name that appeared in both of their family histories, and they chose that one. I thought that was cool. It works better if you are from similar heritages, though!
And I guess ANOTHER option is for HIM to change HIS name to YOURS. I have also known couples who did that. Who says the woman has to take the man's last name?!