By periwinkle, do you mean Vinca minor? How much sun is it in? And don't worry about spiders or their webs - they are almost universally garden beneficials, consuming other damaging insects. They cause no harm to plants.

Any chance of pictures?

Rain shouldn't worry Vinca minor. When you say it might be overwatering does that mean you have actually been watering it? I don't know your climate or weather but it shouldn't really need irrigation unless you're in drought.

Are the webs definitely spiders and not some kind of tenting caterpillar?

on my near yellow beach sand.. i have NEVER water vinca after the year of planting, in 23 years...

we do need a pic.. to insure we are talking about the right plant... periwinkle is a color.. and at least two plants... lets verify the plant..

ken