Sounds really like it it is not getting sufficient fuel.
However give up your obsession with the "runs without " bit
As in the previous post, without a string there is nothing resisting the engine turning apart from its own compression and the resistence of the internals of the shaft.
Once ther is a string hanging out of the head, it has to push it's way through the air so it puts a load on the engine. A longer string applies a bigger load.
The fact that it will run no load suggest that the electrics are OK so that leaves fuel & compression.
Now I am fresh out of fairys to pop over there and put a spell on your trimmer so you will have to do some work yourself.
Do you have a compression tester ?
Do tou have a pressure pump ?
Do you have a vacuum pump ?
Start by going over the the Walbro site
Parts & Service down at the bottom riht are some links to service manuals and tech tips, click on them download the manual and read the tech tips all cube carbs work the same way regardless of brand and there are basically only 2 types the butterfly & the rotary.
Make a coffee or two and read he entire manual at least twice then go back & have a look at your carb.
My initial thoughts are the high speed jet is blocked but go through according to the manual.
It has big clear colour diagrams that can never be bettered by text alone.