Welcome Teresa,
Yes, to get the best geocoding does often need some tweaking to place names.

The reason the Search and Replace plugin does not work is that the Name of a Place record is not a Place field.
That is explained in the plugin Help & Advice under Usage Examples for Place Name Changes that says:
"set Search Scope: Place Records (_PLAC) and tick All other Text fields but clear all the other Basic Filters. That ensures both the Place name and Standardized fields are changed but leaves all other fields unaltered."

It is possible to copy all the Place Names to the Standardized field but will need a small custom Plugin.

Unfortunately, it is not possible for the Search and Replace plugin to operate on just the Standardized field, but it can operate on just the Place Name field by selecting Record Names/Titles Place Name (_PLAC.TEXT).
So the solution is to convert all the Place Names until geocoding is better, copy them to the Standardized field, then convert just the Place Name fields back to your abbreviated format.

An alternative is the Map Life Facts plugin, which can sync with Place records and a few quick experiments with your 'problem' Place names showed that it correctly geocoded Jefferson Co., St. Lawrence Co., etc, in the north of New York state.

BTW: There are some settings you may have overlooked when exporting GEDCOM from Legacy for FH.

Mike Tate