To check your nut height, fret the string at the 4th fret and hold it down then with the other hand press down at the 2nd fret there should be a tiny amount of play, if there is anything more significant you nut needs adjusting.
You can use a credit card as a rough gauge, but as the average credit card is around 0.79 or 1/32” thick so it should be a tight fit, if it slips easily in and out it is too high for most players, unless they want a high action.
You might also want to check your neck relief as too much relief can make the nut action seem too high, and you may find some videos on youtube regarding setup.
But general speaking if you don’t like the way your bass plays I’d take it to a guitar tech, a decent setup makes the world of difference.
BTW dropping the action at the nut rarely means replacing the nut and is an easy job with the right tools, but it is very easy to bugger up if you are not sure what you are doing and do not have the right tools.
Dave.