What is the opposite of "avoidance"
Solution 1:
How about "engaged"?
Last year I avoided some problems, but this year I resolve to be engaged.
Solution 2:
To me, the opposite of avoidance is engagement.
EDIT
A good word in your particular sentence might be diligent. It's not exactly the opposite of avoidance but I think fits better in your construction. Instead of avoiding problems, you want to be aware of them and work to solve them. Perseverance.
Last year I avoided some problems, but this year I resolve to be diligent.
Solution 3:
If a phrasal verb will do: seek out.