When you said you measured the 5V pin you didn't post the voltage you measured so I'll take that to mean you simply verified it was in fact 5V.
First of all, I'd like to say that it appears you have not done any homework on this.
This is one of the first links that pops up on Google with the search "AD590JH SCHEMATIC"
The datasheet shows the third pin unmarked , as it should be since it is the case.
The following circuit should have an output of 2.982 V @ 25 deg C
AD590 AP NOTES
AP NOTE AN-273
NOTE: AN-273 SHOWS AD590 as CURRENT SOURCE driving 1 k /0.1% precision resistor create voltage source (1 uA/deg K)
R = 1000 ohm
I = 298.2 uA @ 25 deg C
V(25 deg C) = I * R = 0.0002982 A * 1000 ohms = 0.2982 V
Vamplified(A = 10) = 0.2982 * 10 = 2.982 V @ 25 deg C
Regardless of how many pins are on the device , the datasheet clearly states it is a TWO terminal device , so the pin you labeled "ADJ" is science fiction.
The AD590 is a 2-terminal integrated circuit temperature transducer
that produces an output current proportional to absolute
temperature. For supply voltages between 4 V and 30 V, the device
acts as a high impedance, constant current regulator passing
1 μA/K. Laser trimming of the chip’s thin-film resistors is used
to calibrate the device to 298.2 μA output at 298.2 K (25°C
I found this schematic which shows the device connected in series with a fixed resistor and a pot with the object of calibrating the device by adjusting the series resistance as close to 10000 ohms as possible.
EVERY SINGLE SCHEMATIC I can find online using Google Images shows the device connected in series with 10 k ohms.
NOT ONE shows the device having a THIRD pin labeled ADJ.
Did you just make that up ?
The schematic you posted is nonsense, since you are trying to read the case.
The question now , is have you damaged the device by connecting it directly to the power supply with no series resistor. The circuit schematic appears to be quite complex so I would not rule out the possibility of short circuit protection. If the current through the device is 1 uA per degree K and 298.2uA @ 25 degrees celsius, and it has some protection, the device would look like an OPEN circuit to the 5 V power supply .
Try connecting it with a 10 k resistor in series first with the resistor end connected to GND and see if the voltage across the resistor is 2.982 V dc. If not , then you probably fried it when you connected it directly across the supply, but somehow I doubt it is damaged, I think you have yet to connect it correctly. We'll know after you try that. Next time try doing some homework before you start messing around with electronics you are not familiar with. Normally , I would be nicer about this but in your case you didn't seem have made an effort to research the subject. You could not have read the datasheet or looked at any schematics online. Where you got your schematic is a mystery. Where did you get it ?