Organisms have many ways to repair DNA. In most cells, DNA damage does not build up, but it is repaired. However, repairing or copying DNA can introduce mutations. And these do accumulate. However, since the emergence of the first double stranded DNA molecule, mutations have been building up and persisting for 3.4 to 4.5 billion years. This is true for every DNA molecule in existence. In slowly dividing cells, DNA damage can be an issue, though.
What sexual reproduction does is merely select a single variation on the DNA and create an entire new organism out of that variation. And in that organism, mutations are free to deviate as there is no mechanism that checks the DNA in our toe stays the same as the DNA in our ear. Same is true for plants.
Now there are many effects in cell biology related to aging. And this can link to DNA in different ways. It is a bit misleading to say that aging is caused just by DNA damage. It is much more than loss of control over biochemical processes that cause aging in humans than actual mutations in DNA. And it being a lot more true for humans than for trees. In general, age is determined more by species than by the age of a species. A 7 year old human will not have aging tissues. But a 7 year old rat will. Cells with unrecoverable bad DNA are killed. And cells where these DNA repair and control mechanisms are damaged become tumor cells and grow more rapidly than the original.
Also, this is all quite well understood.