Allanon didn't have much time in the original trilogy to do anything. He was either trying to restore his vitality with Druid sleep, or using up that essence saving the world. There was no time for a lengthy apprenticeship.
Every one of the Ohmsfords he had to drag kicking and screaming into the world saving biz. None of them would have been willing to set aside their entire life to take up his mantle. Only later generations, long after he died, would do so.
- Lots of the plots are the Druids fault.
Warlock Lord, fallen druid. Ildatch, Druid relic. Setting up the Druid order again, without taking this into account, is a disaster waiting to happen. Allanon can't risk that, and though he needs to pass his guardian role on, he fails to find any way to do so before his sacrifice is needed.
There is another anyway. Cogline, the eccentric old man seen with Kimber Boh in Wishsong, is in the next series revealed as a Druid in exile. Allanon knows if his plans fail he can fall back on Cogline, and in fact does so repeatedly, his shade compelling Cogline to assemble the heroes of the next series and heavily assisting the one intended to be the next Druid.