Yes, Dhanushkodi and Rameswaram are on the same island. No bridge is required to connect them.

There were two old alignments for this line. In the attached screenshot, the orange line with the white bars (on the right side) was the original alignment from Rameswaram to Dhanushkodi, opened in 1914. This line was constantly covered by sand dunes that had to be cleared.
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To avoid this problem, this line was closed and a new line was constructed on the south side of the island from Pamban station, avoiding Rameswaram (The orange line with yellow bars at the bottom of the island). This line was opened in 1927 and was in operation until 1964 when it was damaged in a cyclone and abandoned.

At the place marked by the red circle, a station called 'Rameswaram Road' was opened in 1927 when the new line constructed. This station was too abandoned later.

Now looks like they are going to use the original alignment again (with the white lines), with some change near Rameswaram station due to the expansion of the town.