Sessions are OK
The test worked OK for me so apparently it's something I inadvertently messed up though I can't figure out exactly what I did wrong. The changes were pretty simple.
The remedy is running the Search Wizard 9 times to create 9 different pages & then save the generate code into 9 separate "include" files. Test the $search_type value at the start & include the appropriate code. A bit tedious to set up but apparently safer by far.