Yes you can add a primer and no it will not make the trimmer easier to start.
On diaphram carbs the primmer simply fills the diaphram chamber with fuel, equivalent to filling the float bowl on a normal carb, it does not flood the engine in any way shape or form.
If the trimmer starts ok on the choke then will run on the choke but not when you turn the choke off it is not geting enough fuel for the amount of air getting into the combustion chamber.
Start with checking the mounting bolts are snugged up, don't over tighten most are done up 2 to 5 ft lbs which is screwdriver tight.
Next thing to check is the fuel lines.
On old trimmers the input lines can crack , generally just inside the tank so the trimmer sucks air rather than fuel.
check this by filling the fuel tank to the top of the filler cap,
Problem goes away the time for new fuel lines.
Second test is to use an alternate fuel line, problem goes way = blocked fuel filter.
neither of those then it is either leaking crankcase seals or blocked fuel passages in the carb.
And yes it is easier to just swap them over as a lot of times they do not respond to cleaning and every time you open one up you should replace both the diaphram & fuel pump gaskets and in a lot of cases this means buying a full rebuild kit which is 1/2 to 4/5 the price of a ( hopefully ) new clean carb.