try to remember where the sharps and flats are in the scale and count the intervals while the scale is playing

Well, the thing with scales is that there are so many of them, that you have to have a basis from which to build all the others. This is why we usually refer to the major and natural minor scales for reference. It is much easier to remember that Dorian is a natural minor scale with a natural 6th degree, than whatever tone-tone-semi tone you can come up with (imo).

So with that, you should probably start by trying to recognize what the 3rd degree of the scale is, and then figure out what makes this scale "unique". This approach actually made it hard for me to naturally recognize a natural minor scale, since it doesn't have a "unique characteristic" lol.

As for using your instrument, I think that in a game like this, where you can clearly hear all of the notes, using an instrument is not ideal. However, I highly recommend trying to sing the scale, and trying to notice the different scale degrees like that.

Good luck!