Is my Arduino Ethernet Shield broken?
I'm sorry to tell you but i think you damaged your board severely that's because you took more energy from the pins than an arduino can handle.

150 LEDs takes way too much power for the arduino to handle as you can see here.

So you likely burnt something on your board and might have to replace it :/
make sure which board is damaged(the arduino itself or the shield) before replacing it.

If the ATmega328P(The chip on the arduino) is damaged then this should be an easy fix.

Instead of what you did, you shoud have powered the LED strip from another power supply(not from the arduino pins) and used the arduino to turn "this power supply" on and off(probably using transistors).