Please offer some advice.

VSK has very well outlined the principle of completing campaigns; I was also going to refer to it. Although the details of the walkthroughs (on Champion difficulty) may differ, the general principle remains the same. If you don't prevent the Zombies from taking the artifact on the 3rd day, you'll have to take it on the 10th (when they finally retreat). Units gather resources and artifacts, the Hero takes dwellings, builds up an army, and levels up on upgrade shrines, avoiding battles as much as possible (losses will be disproportionate). The goal is to go and capture the weakest town (in the middle). Employ a Scythian-like tactic: while the enemy captures territory, destroying the guards, you take their objects, buy out the growth in dwellings and in the town, and when the enemy returns, you retreat (but this is after you've captured your first town; otherwise, if you fight and retreat, there's nowhere to go). When enough troops (your own and stolen) have been gathered, you can destroy the barbarian enemy, take all the barbarian towns, and clear the territory. Then, go beyond the borders to destroy the other enemies. Just don't forget to level up your Hero, or he won't survive the final battle (the hammer's attack is too strong). I don't remember how to complete the subsequent scenarios as the Barbarian; the first ones are usually more difficult.