As quiltmum says, the trick is not to get the label wet before your first attempt. You can sometimes dig off the label without leaving the adhesive behind, but often some or all of it will be left. From my experience, most products need two tries if the label is still there: first one to shift the label and second to remove the adhesive, so it's good to get the label off before you start: if it won't dig off, trying soaking it off. I use WD40 sprayed onto kitchen towel to remove the adhesive, moving onto other parts of the towel and re-spraying regularly because the adhesive glues up the part of the towel you were using. I find that washing the jar in my normal washing-up - possibly with a bit of added washing-up liquid spread on the label area - cleans the jar and removes the smell of WD40.
For postage stamps I've removed: if they're still sticky on the back, I store them on the empty backing sheets from address labels or postage stamps I've bought. Wondering if using embossing buddy or cornstarch might make it difficult to adhere them to a project?