Hi Jesse:
People use different methods for accompanying themselves, but I think my approach is both simple and effective.
I start by learning the tune and a suitable set of chords. That gives me a chord sequence, and also gives me confidence that I will have an arrangement that works.
I always want the lyrics of the verses of a song I sing to be absolutely in the forefront of the performance and to be clear - and I don't want my melodeon to distract from them or the tune that my voice is carrying... More than that, I don't want the task of playing a complicated tune to distract me when I'm supposed to be singing. (You may have noticed that there are quite a lot of distracted singers playing a variety of instruments around... )
So while I don't make a rigid rule of this, in the verses most of the time I aim to play sympathetic chords fairly quietly using buttons on both sides of the box. Some tune creeps in quite often, but it's frequently a bit simplicated and follows the singing rather than the other way around (I hope).
I've found it helps to buy a quietish and mellow melodeon for this purpose, but it also that it helps to practise playing quietly. (I think we should all do that anyway for the sake of developing control.)
On choruses, however, the words matter less (the crowd often know them already) and in general we want to raise the roof. In contrast to the verses, therefore, I nearly always play the tune on choruses: it adds to the general drama and noise, and seems to encourage folks to sing along.
This oldish video names and demonstrates the chords I think I'm still using on this one.
https://youtu.be/qzhRXLdzDuQI hope this helps!
Gavin