I think it is important to follow grainlines - because I think a quilt, blanket, sheet, towel, handkerchief - is easier to fold neatly when it is cut on-grain.
First, I wash and dry the fabric to see what it wants to do "on its own."
Then I look carefully to see what I have for grain lines on the ends.
If I have "extra" - I will tear it to get an on-grain edge. Tearing does cause trauma to the ends. Sometimes the damage can extend several inches into the fabric.
If I have to be careful with what I have available - I will pull a thread - which is tedious - but it does work.
I do remove the selvages from backing material. Most selvages are more tightly woven than the rest of the fabric - and some of them shrink a lot more than the rest of the fabric. So they get cut off.
If only a scant amount of fabric is available - then I suppose you will have to "make do" with what you have.