Q:

What does ぐてっと mean? 

I saw it in the following context:

xxは、ぐてっと俺に体を預け、胸に顔を埋めてくる

Based on the spelling, I wonder if it's similar to ぐでんぐでん, but I'm not sure about that.  Based on context, I'm guessing it means something like to collapse (out of exhaustion/tiredness). 

A:

「ぐてっと」 roughly means "in a lazy or lethargic  manner", "lacking energy", "lifelessly", etc. 　「ぐたっと」 and 「ぐたりと」 virtually mean the same thing.  Just think of Gudetama and you will get the picture. 
So, this person kind of leaned against another in such a manner. 
「ぐでんぐでん」 is quite different as it means "in a highly intoxicated manner". 
「ぐてっと」, all by itself, does not suggest any alcohol consumption.