- 扱う: literally "do something with it while holding it in hands"
- 捌く: literally "quickly put away / clean up"
- 解決する: straightforwardly "solve" or "settle"
- 処理する: "process" or "dispose of"
The last two just mean what those words mean, so the problem is the first two. They each carry specific metaphor when used with abstract things.
扱う puts a focus on careful handling, or the method you take when you handle something.
is thus heard when you take care of it wisely without messing it up or letting it break down etc. In this collocation, 扱う also could be interpreted as "bring up (as a topic)".
捌く suggests efficient or high throughput processing. When you say
it is like you deal with it nimbly, with minimal effort, or arranging people to have it done smoothly etc. Sometimes it even implies triaging (i.e. you don't cope with every corner of the problem).