Seems the info in the Traffic Data Report, TDV, and NYSDOT shapefiles just inventory the routes in one primary direction, with one-way pairs not (fully?) acknowledged. Others (vdeane? froggie? Jim?) have probably spent a lot more time with these sources and know more about their conventions and quirks.
It looks like eastbound NY 352 is not carried on Water St all the way to NY17 as shown in the HB (Water/Church couplet), but rather that it jogs north on Madison Ave, then follows Church St to NY17.
Some signage may have changed around 2013-2014, but it's tough to get a complete picture from GMSV, with a lot of imagery still dating from 2012-2013.
I'm betting that the westbound route is still carried on Church St.
vdeane/others, do you have the scoop on where NY352 goes at its east end?