In phonology these are called "minimal pairs," where two words with different meanings are identical except for a single sound change (like "bat" and "cat"). It makes it very easy to illustrate the difference between two phonemes in a language, or to show how people speaking in accents may substitute one sound for another but still be understood.
In this sense, don't get too hung up on the proper pronunciation of the ん, because even if you always pronounce it as "n," it would still be understandable--you would just have a noticeable accent.
Here are some examples. The machine may be a little hard to hear, just remember that whatever the next sound is going to be make, move your mouth to that position and make a nasal sound (voiced and breathing out the nose), and you'll find that the ん doesn't
have a rule, it
describes a natural thing that happens when we speak, called assimilation.
参りましょう / Mairimashou is a very polite way of saying 行きましょう / ikimashou, meaning "let's go" or sometimes "let's get started."