Help / advice needed!!

So I have never let my baby sleep with a bib on however she is now 4 and a half months old, I think she is beginning to teeth and she drools so much, even in her sleep (Which I know is normal) however when she wakes up (she sleeps 10hrs overnight) she has a red rash almost round her neck and chest where her vest/sleep suit and sleeping bag and soaking from saliva. 

what does everyone do to prevent the wetness causing a redness? I don't want to be waking her through the night to change her seeing as her sleep pattern is already so good. 

Any advice would ne appreciated.. TIA