Are there any water based creatures in The Lord of the Rings aside from the Watcher in the Water who attacked the Fellowship in Moria?
Not specifically Lord of the Rings, but definitely Middle Earth; here we go!
The poem Fastitocalon describes a typical myth of sailors landing on the back of a sea creature, and is set in Middle Earth.
The Silmarillion describes great underwater weeds in chapter 3, and the earlier Lost Tales drafts make much reference to Ulmo's "Fishy Car" - so there are definitely salt-water fish in the Lost Tales (Ulmo being the Vala of the oceans).
There are definitely sea gulls - both Tuor and Legolas see and hear them. Looking at their diet, one may reasonably infer that sea creatures most probably exist to form part of it.
...and probably other references to fish that I've missed.
Finally, and given that we have explicit info from Tolkien that Middle Earth is our world, at an earlier stage of its history (see Letters for more on this), and also given that we have plenty of fossil record for sea creatures going back millions of years, we can quite reliably say that sea creatures do exist in Middle Earth.
Going by just The Lord of the Rings, the answer still appears to be "yes": in "The Council of Elrond" (Book 2, chapter II), Glorfindel at one point suggests throwing the Ring into the deeps:
‘... in the Sea it would be safe.’
‘Not safe for ever,’ said Gandalf. ‘There are many things in the deep waters; ...’