There is a very good chance that the side effects will improve.
I was on Avonex for about 10 years, and felt a bit rough to start with (I think most people do), but never too bad. I was soon into a routine that worked for me: inject last thing at night, take a couple of ibuprofen before bed and set another couple out for the morning, take them first thing. The morning dose of ibuprofen was key for me. Remember that (even if I felt I didn’t need it), and I would be OK: mildly hung over for the morning, but OK. Forget that, and I would feel grotty for most of the day. But only a bit - never close to bad enough to have to take time off work or anything. Oh yes - if I did find I wasn’t sleepign too well on injection night, a couple of Kalms (over-the-counter herbal sleep aid - very gentle and effective) would do the trick.
It was a really good drug for me. Avonex kept me well for many years (or I think it is fair to assume that it helped a lot - you never really know with MS, do you?) and I was very glad indeed to be on it. These things take a bit of getting used to, that’s for sure, but for me it was very much worth the effort, and I hope you find the same.