Are MOSFET drains ESD sensitive?

Yes, mosfets drains are sensitive to ESD. If you look at the Vds spec in the datasheet, it cannot be exceeded (for the IRF530, it's 100V), even for short amounts of time.

ESD over-voltage conditions can reach thousands of volts. To prevent failure of the mosfet, install ESD protection. One way to do this would be to use a TVS diode that limits the voltage on the drain to less than 100V (maybe 50V would be best).

If its a choice between placing a TVS diode or replacing the mosfet (or having an intermittent mosfet that behaves unpredictably, a TVS diode would be a better choice.

There are a few mosfets, such as the DMN61D8LQ that have ESD/diode protection built in to the part, however, there might not be one with the specs needed for your design. If needed TVS diodes could be placed on the outside of a mosfet to give it ESD protection or additional ESD protection (if the ESD rating of a mosfet isn't high enough).

Source: https://www.diodes.com/assets/Datasheets/DMN61D8LQ.pdf

If the LED's are connected to the outside world and are touchable by humans (or moving cables can generate thousands of volts on them if moving across a surface), then the mosfets need to be protected.