I've used Avonex for a very long time now but what you are saying is what I always assumed would happen if I didn't do something else as well. I'm sure your neurologist is well meaning I suspect he/she doesn't really know how or why Avonex influences MS. You will probably get an answer if you ask but you probably won't be able to judge the quality of what your told anyway so lets think of some other things.
"Wierd" is how I feel when I don't manage my MS so I think it is a very good word to describe how you feel.
At a bare minimum, a set of tests would show you have a very low level of Uric acid, low levels of B vitamins, low levels of some non essential amino acids , impaired Vit D and and low readings on some minerals. The problem is Avonex was never designed to deal with these things on its own. Interferon is a signalling agent to alert the immune system and that is all it really does. If you don't deal with underlying issues eventually your immune system is too exhausted and the Avonex becomes ineffective. I'd say the Avonex was working but your system has become exhausted. Your neurologists approach just sounds like he's stabbing around in the dark and if he hasn't done any of the tests to check those levels then you might as well take advice from a next door neighbour.
There are several different approaches people take on these boards. I prefer to assume that underlying cause is an EBV infection and treat it accordingly with great success. (see Avonex and Valtrex under the regimen section). Others have different approaches and there are common elements but different ideas as causes.