I have had some luck with the syringes too, mostly on chairs. They
make kits with a thinner glue that supposedly spreads out better
within the joint. But, I have done OK using regular carpenter's glue
BTW, the glue syringes have varying sized of needles, and don't try to
just poke it into the wood. You should work the needle into the edge
of the joint, or drill a small hole just larger than the needle.
Don't need a alcohol swab either.