Yes, use ArcPy. The tricky part is to only list the feature classes and fields you want to change.
This example will list all feature classes in the workspace that startswith Testnull. All string/text fields will then be listed and None values replaced with empty strings.
Make sure to backup your data Before you try it. Modify and execute in the python window of ArcMap.
arcpy.env.workspace = r'C:\Default.gdb' #Change to your database
featureclasses = [feature for feature in arcpy.ListFeatureClasses() if feature.startswith('Testnull')]
#featureclasses = arcpy.ListFeatureClasses() #Use this line to list all feature classes in workspace
for fc in featureclasses:
fields = [f.name for f in arcpy.ListFields(fc) if f.type=='String']
with arcpy.da.UpdateCursor(fc,fields) as cursor:
for row in cursor:
row = ["" if v is None else v for v in row] #If value is none, replace with empty string, else reuse the value