I am pretty sure it is. It is discussed on Wikipedia, but I am sure if you Googled around you will find similar discussions. That is why you don’t hear much about Dolly or the glories of cloning any more.
Bottom line is that anything cloned from a host will be the same age. So forget about extending your life with spare body parts.
I actually did google and didn’t find anything that confirms your assertion with the exception of people like you speculating about it.
Wikipedia even says
“There were early claims that Dolly the Sheep had pathologies resembling accelerated aging. Scientists speculated that Dolly’s death in 2003 was related to the shortening of telomeres, DNA-protein complexes that protect the end of linear chromosomes. However, other researchers, including Ian Wilmut who led the team that successfully cloned Dolly, argue that Dolly’s early death due to respiratory infection was unrelated to deficiencies with the cloning process.”