If I understand you correctly, you want to combine polygons that are within a specified distance into larger polygons ONLY IF they share a common field value. You can try the Select Layer By Attribute followed by the Aggregate Polygons tool (ArcInfo license; Cartography toolbox - Generalization toolset).
If you have many unique field values for selecting features, you can build a model using the Iterate Feature Selection iterator (inside ModelBuilder - Insert - Iterators ...); specify the field you want to iterate as Group By Field (making a selection per iteration). Then use the Aggregate Polygons tool to combine/merge the selected polygons within the desired distance.