Adam, the DC-DC converter listing says 'up to 1.4A'. Allowing for that being the absolute maximum output current, you can bet that's at the absolute minimum output voltage. ie 2.5V
So, 2.5 x 1.4 = 3.5W
3.5W / 12V = 290mA
All of these sellers tend to exaggerate, so it probably can't really supply this much current continuously.
Then, if we take the monitor seller's claim of <3W, and allow for some exaggeration here as well, this power supply is probably insufficient, or at the very least borderline. (In the listing, they also say that it uses 120mA.)
To be on the safe side, I'd go with something that can produce at least 0.5A at 12VDC.
jasongitar, as suggested by Skidood, at this point I'd test the monitor on a good 12VDC supply, perhaps a 12V SLA or other lead-acid battery to make sure it works OK, measure the current if possible, then get a better-suited 5V to 12V DC-DC converter. (The second one that you linked to and purchased might still be a bit on the small side. You can bet that the maximum 3A is probably only available at the minimum 1.25V, making 3.75W total. 3.75 / 12=313mA - pretty close to the line still.)
As Skidood also said, if the original DC-DC converter has released it's magic smoke, it might well be dead now.
Test it for 12VDC out, then if it's OK put it aside for another project.