Are there hot melt glue sticks suitable for use on a PCB which do not make a lot of spider web like strings?

 

Is that like non-stick Duct-Tape ?

The quality of your Glue-Gun, and the cleanliness of the Tip will be a good start,
then You need to work on your technique.
Disposable Terry-Cloth-Rags are great for cleaning the Tip.
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There are “low stringing” glue formulations but for whatever reason they only seen to come in pellet for for use with melting pots (or electric skillets as is often the case) rather than stick form.

In industrial applications the usual advice is to ensure there is sufficient space between the nozzle and the surface. We usually apply it in contact with the bead in manual work, but hovering a bit can help.

Another thing that can influence stringing is temperature. If the temperature of the gun is too low, the glue will start to harden before it has dropped away from the nozzle. If the work is too cold, a similar thing can happen as it quickly cools the glue applied.

The last thing is the state of the nozzle. If there is char on it, or accumulated degraded glue, it can retain the little bits that be one strings. Clean or replace it if you can see that.

That said, holt melt adhesives are not all one thing. There are many formulations and some will exhibit more stringing than others. Testing various brands for comparative stringing is probably a good idea. Also ensure you have a good quality gun designed for the temperature of the glue sticks you are using. Too low can cause stringing and poor adhesion, to high can cause dripping and runout.