Depends on what the diseases are but in general, yes, any new trees propagated by cuttings or grafts from the old tree would also carry the same diseases susceptibility of the parent tree. Particularly something like leaf curl.

Better to grow a kind which is resistant to the common diseases in your region.

Are resistant varieties usually gmo or?

No fruit trees that I am aware of (except maybe mangos) are GMO'd. And not sure they are available to the general public anyway.

Most crops that are GMO'd are only for mega agribusiness, not for home gardeners. And there are not all that many GMO'd crops anyway, in the grand scheme of things..

This is one of the issues with the peach tree. It's all over the tree

That's lichen.......and it is harmless :-)

How do I find which variety I should be planting?

I sprayed the tree for peach scab. Thats the only disease I remember for sure it has had in the past years. I used a broad spectrum spray this year on the branches I can reach.

Peach scab? Think you're "remembering" the wrong name for whatever it was.

@Jean - https://extension.psu.edu/peach-disease-scab

This is another disease it has. I don't know what it is