Now, I’m not sure if maybe it is just a level issue, this I doubt seeing as I just tamed a second cat losing all my training progress. Pets seem to not know their natural abilities nor do they seem to learn them as they grow.
For instance, my originally trained cat at lvl10 knew 0 natural abilities. Leveled him from like 8 all the way to 18 and he still learned no natural abilities (claw, cower, bite, etc.).
I know this is not a level issue as I just tamed a new lvl17 cat in Darkshore and he only knows cower. Nothing else.
https:/ /www.wow-petopia.com /classic/ (remove spaces). You have to stable your pet and go tame a temp pet with abilities, fight stuff with the temp pet and hope it teaches you the ability you are looking for…
Claw 3: Pet Level 16, Cost 7 TP. Claw the enemy, causing 12 to 16 damage.Can be learned by taming:
• Den Mother (Bear, 18-19, Darkshore)
• Ghost Saber (Cat, 19-20, Darkshore)
Since you mentioned darkshore, check for those and they will have claw 3 (maybe), if not tame another and so on 
Classic pets do not learn offensive abilities “naturally” or from pet trainers. You have to teach them.
You do realise that you need to train other pets in the wild to learn skills that you then teach to your pet? Such as you need to train a spider or wolf to learn bite, charge is learned from boars, and claw from bears. You stable your pet, tame another animal and make it work until you learn the skill, then abandon it. Recall your pet and you can train it. Resistances are taught by pet trainers after reaching level 20.
Go to Petopia to view what animals in the wild have which abilities.