@Lukeott … the post below shows some examples of propagating raspberries from root shoots and tip rooting.
If the blackberry varieties you have are prone to producing root shoots… and you have had them planted 2-3 years… you should be seeing evidence ot that… the roots simply run out 2 3 4 5 ft from the initial crown and send up a shoot.
Those shoots can normally simply be dug up / separated from the mother crown… and transplanted in another location.
You may have to take care of them well, baby them a bit for a few weeks after digging them up… give them some time in indirect light… then part sun (morning sun)… before planting them out in a full sun location.
I have 6 of the older variety… illini hardy… and can testify to what Kris said… they simply do not root shoot… they do send up new canes very near the mother crown… but in over 20 years that I have had them… never once had a shoot come up more than 3-4 inches from the mother crown.
Most of the modern day varieties do send up root shoots quite some distance from the mother crown. I have some Ouachita blackberries and they will send up root shoots a good 3 4 5 ft from the mother crown.
Very easy to simply dig those up and … perhaps baby them a bit… and then plant anywhere you want.
Good luck to you !!
TNHunter