nancy - Suntour actually offered an AG freewheel with a 38 tooth low and made an AG dérailleur to match... but these are like hen's teeth.
The Huret Duopar was the only other dérailleur that could handle this extreme gearing range.
Most older Suntour GT models will handle 34 teeth if the bike is equipped with a long enough hangar and these longer hangars were also available to increase the range of many dérailleur.
The Vx Gt is a beautiful piece of work and it is what I use on my touring bike that runs a 34 low and it handles this with no troubles at all from the middle and granny... but you don't want to cross chain this bike into the big / big combo.
For the OP - Your dérailleur is a Titlist DB600 which is identical to a 600 series model 6100 and a fine dérailleur for shifting as many as 28 teeth but it won't handle a wide range like a Suntour GT or Shimano GS dérailleur... the Suntour is superior to the Shimano of this era.
A modern long cage mtb derailleur will handle 34 teeth and can be run on friction... these are easy to get while some vintage bits take a little more searching for.
You will probably want a lower gear than what a 42/34 would offer so fitting a triple with a smaller granny ring would really expand your gearing range.