Is solving cancer required in order to avoid aging?
Interesting question. My answer is no, but it requires a rather science-fiction style answer - at least it's beyond current technology, but here goes:

My Assumptions

I make the simplifying assumption that ageing is only related to telomere length. Thus by "avoid ageing" I assume you mean "avoid telomere shortening". Also to clarify things for others, I'll clear up the role of telomeres in aging and cancer:

Telomeres in cancer - it has been hypothesised that telomeres act as "mitotic clocks" - keeping track of how many times a cell can divide before it stops. This "time limit" is supposed to put a halt to cancerous cells before becoming malignant. 

Telomeres in aging - Since telomeres set a limit to the number of times a cell can divide this essentially sets a time limit for the organism as a whole - those cells that divide faster (e.g. hair and skin cells) go out first (grey hair and wrinkles) etc.

Short answer

No.

Long answer

All we need to do is come up with a way to somehow "maintain" average telomere length at some "safe" length, i.e. we keep them at a length that is long enough to prevent us from aging but short enough so that a cancerous cell will not be able to proliferate to dangerous numbers. 

For example, every now and then we provide "telomere length booster jabs" (this is some abstract fictional idea of course). This is like adding more time to our lives when we are about to run out: "Oh look, I only have three months left!", "Don't you worry, with our new telomere booster jab you will have an extra 3 months!"...