Well, I found a super inexpensive solution in the form of a hair color applicator bottle that I kept in my laundry room. It can be found at your local Sally Beauty Supply store for less than $2.
I've started teaching my son how to do laundry and I have it filled with white vinegar (I use that instead of fabric softener and also make my own laundry detergent). The other day I took a closer look at that tip and thought it was narrow enough that it might work with glue.
I was planning on filling up the color applicator bottle with the glue but check this out! It's a perfect fit!
AND it dispenses glue in a very thin line too! (Don't let the photo fool you, it came out in a perfectly thin line but it started to bead up from me walking around with it and trying to find better lighting to photograph it.)
The important thing to remember with this applicator tip is that you need to put something in it when it's not in use so the glue doesn't dry out. I tried a few different size pins but the best thing that seemed to work was a large paper clip.
UPDATE: So I've had the paper clip in for a week and it is NOT a good option. The metal oxidizes and then your glue comes out brown. So I've switched it to a toothpick but a toothpick is a bit too fat for the small hole. I think it will do for now but I'll probably take some sand paper to it to slim it down even further so it fits nicer in the tip.