"But he followed" and "But he continued" are both accepted. I would like to know which one is more accurate.
"But he continued" is more accurate. You'd use this phrase for example when saying "We asked him to stop, but he continued".
Is the verb sequir used more than continuar = continue. I ask because it's just easier to remember
It's seguir not sequir. Por ejemplo: Es difficil seguir sus explicaciones. (It's difficult to follow his explanations. o ?Nos sigue este perro? (Is this dog following us?) These examples from Larousse All Spanish Verbs from A to Z.
Whats wrong with "He kept on going"? Any ideas?
You're missing the word "but" (pero).