I use Betaseron, not Avonex, but they're both interferons. Yes, MS is an autoimmune disease, but that doesn't really have any bearing on colds, as it is not a matter of you having a lowered autoimmune response, just one that doesn't work properly when it encounters myelin.
If you get a fever with your cold (or flu, or other virus), the raising of your body temperature could cause your MS symptoms to worsen for the duration of the fever. But once you get your fever under control. you should be fine. And of course a cold will make you feel miserable, but that in itself should not be a reason to not take your shot.
I hope you feel better soon! Lots of fluids, and rest as much as you can.
...I am not a doctor, nor health professional, and don't pretend to be one, here.....