I should mention, I finally got around to pulling the carburetor on each unit, and removing the screen inside of the carburetor. Sure enough, these screens are really restrictive. Holding it up to the sky, I couldn't see light! But they didn't appear to be clogged with debris.
Anyway, I was able to start the string trimmer with relative ease, and without any starting fluid after reassembly.
However, the blower would not start at all, which was odd....previously, it was the easier starting of the two units!
Both units use a Walbro carburetor. The only difference is that the fuel from the blower's tank has to be drawn out upwards, and the fuel from the string trimmer is pretty much a gravity feed. So the fuel is running in opposite directions for each unit. I am not sure how the blower achieves such a strong vacuum to suck the fuel out of the tank (but it worked for me, previously, just won't work now)
Funny thing, about a week and a half ago I found a Craftsman 22 cc blower in the garbage that is nearly 100% identical to my Weed Eater. Instead of a hand-throttle, it has a combination valve that controls both the choke and the throttle.