DNA damage theory of aging and sex

Yes and no. Your concept about DNA damage causing aging is good. But aging is caused by DNA damage that causes the biologic / signaling pathways to not work as well as they once did. Think of biophysics and interactions among the plethora of proteins, molecules, etc. in your cells and tissues and all of the interactions therewith where even the electronic charge + vs - and quantity of molecules (charged or not) in a vicinity influences the effectiveness of a biological pathway etc. This is aging. Basically, if you have learned about development biology at the molecular and cellular levels, you would know that much of that is not yet developed at the gamete stage. Therefore, there's not a whole lot of aging that can be had at the gamete stage because there's frankly nothing going on just DNA damage. Granted the DNA damage can influence on a grander scale as Bryan points out above - life or death (rather non-life) or the imposition of birth defects and other serious issues - but this is not how we define aging even if they are somewhat conceptually related. In short, aging is caused by the inefficiency and impairment of otherwise optimal (at birth) biological processes that slowly decay over time (silent mutations, environmental impositions, etc. that cause biophysics-based alterations at the smallest scale that impact exponentially as magnified over multiples of molecular interactions across a grand array of biological pathways and other molecular/cellular interactions).

I've answered a related question, which was wondering why older people get more cancer from accumulated DNA damage and yet their children aren't immediately susceptible to it, here:

https://medicalsciences.stackexchange.com/questions/19915/why-do-age-related-dna-transcription-errors-not-manifest-in-children-as-readily/

Quoting from my answer there:

...a parent does not pass on all their mutations in all their cells, they pass on just one, which need not have any deleterious mutations at all, even if the parent has cancer themselves in some other part of the body.

If that one cell is defective in some way, it is less likely to be involved in creation of a successful pregnancy, and may lead to miscarriage. Therefore, the probabilistic process of reproduction tends to result in offspring from selection of healthier gametes.

There are in fact mutations that accumulate in gametes, but if any of those mutations are very serious, they are unlikely to support development of an entire new organism, and they are also relatively few. The burden of age on the genome is spread over all the cells in a body, and worst in types of cells that divide the most, when errors are most likely to occur.