Rather than Vaseline use an olive oil based ointment. Wash and thoroughly dry the whole area, the apply a very thin layer of barrier cream. What you are aiming to do is waterproof the skin not moisturise it. A thick layer encourages sweating, a thin barrier layer allows the saliva to evaporate without macerating/soaking the skin.
Dribbling can often be a sign of enlarged adenoids and glue ear. The mild inflammation narrows the naso pharynx so their snot can’t drain backwards and they often have constant snail trails that they lick and rub. It also makes swallowing uncomfortable so they prefer not to.
My DS had intermittent glue ear as a toddler. When he had grommets fitted and his adenoids removed it stopped completely over night.
Imagine having a sore throat permanently, it makes it difficult to swallow.
Do they still have a dummy? Not a judgement question but it will encourage saliva flow and at three dummy hygiene tends to be lax so the bacteria and fungal spores build up on the dummy causing infection.
Other possible causes include contact dermatitis from wipes or creams already in use. Use a clean wipe each time rather than reusing a cloth because each time you wipe the infection is transferred to the cloth then transferred back when used next. And there is also the possibility of a food allergy. Trial and error is needed to see if it’s triggered by specific contact with food.
DS had a reaction to cows milk early in. Instant rash on contact so we stopped dairy products, waited a while then reintroduced them gradually until it was clear he was no longer sensitive. He was exclusively bf but I don’t like milk so probably wasn’t exposed to it early on so reacted quite strongly when introduced cheese. He’s still not a big milk drinker.
Latex is now a common allergen and more and more people are sensitive to SLS a foaming agent present in many cleaning products including toothpaste. Look for toothpastes with no SLS but still have fluoride.
As a dentist I treat lots of similar rashes. They are usually associated with saliva escaping from the mouth and respond well to correctly used barrier creams.