There's lots of different FETs for different purposes. You should read the number on it and get the same type, however, your problems could run deeper than one blown FET. The FETs are electronic switches, and you have several in parallel. They're switched on when they get about 12v on their left leg. The middle and right legs are the switch, so you should check continuity between those legs only after checking that you have 12v on the left leg. If there's no 12v, there should be no continuity. Therefore, the fact that you have continuity on a FET would probably mean that it's OK. It's the ones without continuity that are the problem. A BMS is different to a controller. In a BMS, the FETs should be switched on all the time the battery is healthy. In a controller, they give pulses of power to the motor, so they're normally switched off.
To test the 12v (could be anything from 8v to 18v), put your black probe on B- and your red probe on the left leg. To decide which is left, you look at the FET from the front with the legs downwards, like shown in my photo of the complete pcb above.
If you look at the PCB with the FETs at the top of the photo, the right hand one is the charging one that opens to allow charging, and closes when your cells are up to 4.2v. The other four are the discharge ones that all work in parallel.
If one were blown, the others should still switch on and allow discharge. If you haven't got any discharge, the transistor that provides the 12v to them may also be blown. Also, water in the BMS could have damaged anything. Try removing the blown FET and see if the BMS works without it. If that doesn't work, it might be easier to get a new BMS.
While you're waiting for a new BMS, as a temporary measure, you can bypass the output FETs by using a bit of 14g wire to bridge between B- and P-. That would only work if the charging FET is still operational. You can bypass the charging FET too by bridging its middle and right leg, but then you have no protection. The charger will cut off at 42v, so no immediate danger, but it's 50/50 whether the balancing is still working. Again, you'd be OK for a few cycles. If you do bypass the output FETs, don't forget that the battery won't switch off when it gets too low. Hopefully, your controller will switch off instead, but just be careful in case it doesn't. Don't let the voltage go too far down. Your LCD should show that.