when we see a change in an animal or plant from generation to generation, a new body part or trait, how do we know if new genetic information has added to its gene sequence or whether the genetic information was always there to begin with and just manifested itself for some reason in one offspring?
Mostly we don't.
Although in recent years it's become possible to sequence nucleic acids fairly quickly and cheaply, so if the funding and time are there to do it, a comparison between the genomes can be made, and the changes investigated.
My understanding is (and I am probably waaaay out of date, as I haven't been trying to keep up, and Molecular Biology has been moving very fast) that a lot of phenotypic change is epigenetic - that is, mediated not by the genetic material itself changing, but by changes in what genetic material is read.
Methylation was getting a lot of press a while back - a gene can be turned on or off by the removal or addition of methyl groups, without a change in the underlying sequence of base pairs.
Ultimately the answer to any question in biology, and certainly any question in molecular biology, is "it's more complicated than you think it's going to be, even after you take into account the fact that it's more complicated than you think it's going to be".
The popular narrative of a simple string of base pairs, in three base groups each coding for a specific amino acid, which is then assembled into a protein, with the changing of one base pair thereby having the potential to change the properties of the final protein, and thereby ultimately the phenotype of the individual in which that change occurs, is not
untrue, but it's such a dramatic oversimplification of the hyper-complex reality that on a clear day, you can see 'untrue' from there, if you squint a little.