As far as I know, the primary danger of disease spread in plants is not actually disease...diseases are fungal, not bacterial or virus.
Most diseases I know of that are spread by plant parts are either on the surface of seeds or are on actual plant parts...leaves, stems, etc. The "diseases" that are spread by seed are less likely to be fungus and more likely to be virus.
If you are not crossing international borders, I'd not be too concerned. If there are no quarantine zones where you are or are moving to, specifically for what you mention, either the plants or the diseases, I'd not be too concerned.
There's a relatively simple
hot water treatment for seeds (122 degrees for 25 minutes) to "clean" the seed of anything that might be catching a ride on the surface...seeds are plant parts, and while very few diseases are actually in the seed itself, they may well try to catch a ride on the surface of the seed coat or shell.