If I remember there was a signal box on the south end of the Tay bridge which would have been the junction for the branch to Newport and Tayport. St fort station masters house was where the A92 crossed over the railway near a hotel now closed.
If you look at an ordnance survey map clues to the Newburgh line seem to follow south of A92 to kilmany then westwards to north of A92 before crossing again at Luthrie then following A92 to Parbroath crossroads. Then south of A913 to Lindores then Newburgh.
The line is fairly easy to follow as cuttings and embankments still exist some brickwork where bridges have been dismantled and then made safe by building walls on the top.
In the 60's I used to visit all the stations in the Dundee area and I cannot remember of the esplanade being used, always looked unsafe to me in a poor state of repair. It was built onto the main line, old timers in the railway told me that that was the old Caledonia line but nor sure if this is Correct, maybe Bill will correct?