As far as I know the TnF was not designed for anything but liquid solvents (cements). There are CA applicators that have a needle tip, but the TnF isn't one of them. You'd never be able to clean it out and the CA (as you probably discovered) sets up inside the glass tube.
For CA applicators I use Teflon tubing inserted into the end of the bottle applicator. It gives me far greater control and I can reach into tight spots to place a dab where I want it.
CA's viscosity (even thin) is too thick for such a small opening that the TnF needle applicator uses. The needle applicators I've used had a piece of wire stuck down them to keep the tunnel clear. They are usually made of a metal that CA has a hard time latching onto.
I don't have my TnF instructions, I've been using them since they first appeared, can't remember if it said anything about using CA or not.