Generally if you can't find the part, you'd have to make it yourself. There are many tutorials explaining how to make a part in Eagle.
You are correct in discerning between the three categories for an Eagle part.
Schematic symbol:
This is the representation of your part as it will appear in the schematic. It doesn't have to have all the pins represented exclusively, like say if there are multiple GND pins.
Footprint/package:
This is what the devices physically looks like and how it will appear on your circuit board. Each solderable pin must be represented here. You need this so you can actually place the part on your PCB.
Device:
This is the middle ground between the two, while you never place a device, it is how the schematic symbol will connect to the foot print in the software. You need this because a connection made in a schematic represents a physical connection on the circuit board. If you connect a signal to a pin of a symbol in the schematic editor, the software needs to know to make the same connection on the circuit board.
You'll notice, after you've made a schematic that has connections, that when you start to design your circuit board, you'll see many yellow lines, these are called "air wires" and are un-routed representations of the signals created in the schematic. In the PCB editor, you are just making the connections you made in the schematic editor a physical track that will be on your circuit board when you have it fabricated.
For this part specifically, I know this has been made and released since I use it all the time and I didn't make my own. Sparkfun releases many of the parts used in their products in the form of the footprint and schematic symbol. It seems like they have a DIL28 connector in their connections library. You can download them from the linked github page.
As for more Eagle library resources:
Best Answer
I can think of three options:
Zoom out as much as you can then use the route tool on the tiny board, this catches the air wire, then zoom in again and route it.
You can also disable the top and bottom layers so the air wire becomes more visible.
Yet another option is to run the provided "length.ulp" script (File->Run... or ULP button). This script shows a list of all the nets, on that list there is a column "Unrouted", some net is not completely routed a value should appear here instead of "--". You can then type on the command line "show net_name" to highlight it.