Yes, I believe you have a strand of freshwater pearls. You got a very good deal!
You didn't post a photo of the bracelet. Do the pearls look the same, have the same clasp /same clasp markings? If so they may have been sold as a set originally.
They could do with a bath and restringing. It's not hard to do and would save you money not having to pay anyone to do it.
To wash them, put some distilled water in a bowl and swish a bit of mild soap in it. Let the pearls sit in the solution for 15 minutes or so, then wipe them clean with a soft cloth and rinse them in some more distilled water. Lay them to dry on a soft towel for 24 hours.
Have a look at my restringing tutorial:
How to string (or restring) your pearls on doubled Pattye's Serafil, Beaders Secret, and Power Pro
This is not hard-- really! It is harder to describe it than to do it. Anyone can learn to string their own pearls!
If after reading through this, it sounds too daunting, I suggest making a mini-strand for practice first. Make one 12-15 pearls long just so you can practice how to knot the end 3 pearls on each side and attach the clasps; the middle pearls are easy to knot.
To just practice knotting (without bothering with clasps), make an overhand knot, string on a pearl (or...
Solid nacre FWP are becoming scarcer as the Chinese farmers are now bead nucleating virtually all FWP.
