You could also try the Dollar Store for super glue.  They have a "2" pack small tube for a buck.  It's the thin stuff and if you use the applicator Alfakilo sugggested or make your own from a large needle. Saves the glue and makes application very easy. 
Best of luck in finding your source.

I clicked on this thread 1 year after the last post to be enlightened about glue. The name Amazon seemed to figure prominently as a source. At that, I entirely lost interest. If the choice between buying from them or mixing up dog manure with fingernail polish to make my own glue, I'll choose the latter. I don't buy from oligarchs. i don't need ANYTHING that much.

Here's the US link:

https://www.amazon.com/Gorilla-Super-Glue-Gram-Clear/dp/B00OAAUAX8/ref=pd_bxgy_i...

Good question! I don't know of a Gorilla CA Glue with a medium thickness, seems to be the gel or the typical very thin type that can get messy as you said.

Perhaps the way to help you is to deal with how the thin type CA glue is applied. My experience is that these thin glues are unpredictable in how the glue comes out of the dispenser. Sometimes a trickle, sometimes a flood!

What works for me and others too is to apply the CA with a pointed metal tool. One solution is to cut the eye of a large sewing needle in half, leaving an end that has a channel to hold a small amount of thin CA thru capillary action. Or there are commercial products specifically designed for this. I've tried both and now use the commercial tool to apply very small amounts of CA, for example when building a box fuselage out of 1/16" sticks.

I dribble 2 or 3 drops of CA into a plastic lid or cup, and then dip the needle or tool into the glue to get a tiny amount for application. Since CA does not "dry" in the traditional sense, the CA in the cup stays thin. For a cup, I cut the bottom off of a small empty CA 'bottle' (see picture).