Electronic – Use MOSFET series resistor instead of shunt for current measurement

It sounds like you need a heatsink.

Another thing to check is your gate voltage when you have the FET switched on. If your Gate-Drain voltage is too high, the MOSFET may not be properly biased on, a situation which would generate a lot of heat.

Also, how often is this switching? Is it a steady-state thing, or is this part of a switching supply? If it's a switching supply, you will also need to look at the rate at which the system switches.

Anyways, assuming you have everything biased properly (probably a safe assumption, but measure it anyways):

R_DS(on) = 130mΩ @ G-D voltage of 5V

So, with 130 mΩ in series with 3A:

$$V = 0.130 * 3$$ $$V = 0.39$$ $$Power = V * A$$ $$Power = 0.39 * 3$$ $$Power = 1.17W$$

So you're going to be dissipating 1.17W of power in the MOSFET, in the best-case situation.
That will get very toasty without a heatsink. If you're just running this as a bare TO-220 device, it getting extremely hot isn't too suprising.

So, assuming we have a TO-220 in free-air:

TO-220 junction-to-air thermal to ambient equals 62.5 degree per watt.

(From here)

Therefore:

$$Δ°C = 62.5 * 1.17$$

$$Δ°C = 73.125$$ $$Device Temperature °C = 73.125 + Ambient$$ $$Device Temperature °C = 98.125$$

So assuming ideal thermal dissipation on a bare TO-220, it's still going to easily reach ~100°C.
Any environmental factors that further reduce the device's cooling will make it worse.

The IRFB7434 is a rather modern MOSFET, possibly employing trench gate structures. In linear operation, these modern MOSFETs can exhibit a positive temperature coefficient on a local scale, leading to current localization followed by local overheating and finally by failure. Further information on these effects can be found in the application note "Automotive MOSFETs in Linear Applications: Thermal Instability" by Infineon.

All SOA diagrams I have encountered so far in datasheets do not take this effect into account and are thus invalid for describing linear operation behavior. This localization effect also renders thermal models invalid if these are based on the thermal impedance graph, as a homogeneous power loss density within the silicon die is assumed.

It is important to note that the SOA data is not actually measured, it is calculated from models and statistical data. Apparently, the models used by most semiconductor manufacturers are somewhat outdated.

Long story short: You cannot use the SOA diagram of modern MOSFETs or the thermal impedance diagram to predict their behavior in linear operation. Maybe you can modify your circuit to prevent linear operation.