Hi Rebecca!
Actually I like that name a lot! A have a few friends named Rebecca so maybe I am biased haha. But to me it doesn’t sound like an old fashioned name. It sounds youthful and pretty but not immature. And you also have a few options for shortening it: Becca, Becs, Becky etc. but I personally like Rebecca without shortening it.
According to a few websites I found the meaning is something like to tie, to bind, to snare. Here is a direct quote from one website
“The name means "a snare" in Hebrew. In Hebrew the meaning of the name Rebecca is: Captivating or a knotted cord. Other suggested meanings of the name are "snare", "noose", "tied up", "secured", and even "beautifully ensnaring".”
I like the name, but of course it’s your name and if you aren’t comfortable with it then feel free to change it! There’s lots of websites in English with suggestions for names like behindthename, and babynames. That kind of thing. So you can find lots of interesting names there along with their meanings. But if you do decide to change your name, it is a good idea just to check with a few native speakers first just so you can get the correct pronunciation (if it’s a bit of a unique name), and also they are more likely to know of something is old fashioned or has a bad connotation or something like that.
All the best!
Anna