You don't need to trim anything. I use Elmer's all the time, both for piecing blocks and basting the quilt sandwich. I don't dilute the glue, and it works beautifully for both.
I use a bottle with a thin tip for piecing and run a very thin line like you did, close to the edge, and then hit it with a hot iron for a couple seconds. Quick and accurate piecing!
If you should make a mistake and line up the edges crooked, just carefully pull them apart and try again.
For basting, I use a bottle that I got at Sally Beauty supply and snipped the tip off so that the hole's a little bit bigger than the one on the glue bottle, and drizzle it on.
When it's dry, the needle goes through like butter; there's no problem at all with those stiff edges or spots, and it all washes completely out when the quilt is done.