Cambridge Dictionary gives a good specific definition of the verb form, along with many examples (emphasis mine):
PRESENT PARTICIPLE OF arise FORMAL OR LITERARY
This explains why there is no common carotid artery on that side, with the internal and external carotid vessels arising directly from the anomalous arch.
The arachnoid cyst is arising from the middle fossa and compressing the cerebral hemisphere.
However, it is considerably more than what publishers typically fear as an ad hoc collection of diverse papers arising from a conference.
Once again, trauma arising in each of the six age categories was insignificantly associated with meaning.
Small chestnut ocular blotches, and two rather long hairs arising on the surface near these blotches.
Individual variation in child vocabulary competence might best be understood as arising within a nexus of contextual factors both proximal and distal to the child.
As you can see, it takes the form of NOUN + VERB + PREPOSITION. (Possibly also with an adverb.)
Although from seems to be the common preposition that follows it, other prepositions can also be used.
It's possible for the verb to come first in certain constructions:
Arising from external loads were negligible stresses.
However, while not wrong, it sounds slightly strange.
The most closely synonymous wording I can think of that uses an actual adjective would be:
The raised stresses are negligible
Some present participles are used adjectivally:
The cottage had running water.
The wind blew the falling leaves.
She comforted her crying daughter.
However, other present participles don't work as adjectives:
✔ I took pleasure in delighting my children.
✘ I had delighting children.
Most often, we don't use a present participle as an adjective if there is an actual adjective that serves the same purpose:
✔ I had delightful children.
Incidentally, I got the delighting / delightful idea from the ELU question "Can any verb's present and past participles be used as adjectives?"
To me, arising doesn't sound like a present participle that would normally be used as an adjective.
If I weren't going to use the normal verb form and was looking for an adjective, and if I didn't want to use raised stresses, I might say the following instead:
The rising stresses are negligible.
This sounds more natural, simply because I'm used to hearing about rising tides and rising sea levels.
However, there is also nothing wrong with using existing as an adjective as you suggest:
The existing stresses are negligible.
It depends on if you want to talk about something that exists fully formed or about something that is in the process of forming.
On the other hand, I can't say with certainty that arising can't be used as an adjective. It seems to exist in a kind of grey area. But at least stylistically, it's probably correct to say that you should try to find some other wording—especially if the person who told you to is somebody you should be listening to.