Get a “Level too low” message when trying to Techcopy

Are hitting the "x" button at the right time? If you get the timing wrong, you won't be able to get the tech, regardless of levels.

I think you've misunderstood the chronology.

Sin was originally created 1,000 years prior to the events of FFX, and has been defeated several times over the course of that millennium. Each time, there is a period of Calm (with a capital C) before Sin returns.

The Calm between Sins isn't given a fixed period as far as I can remember (or tell from other resources). Sin is most certainly returning around the time FFX's story begins, and it's only been around 10 years since he was last defeated.

To understand Tidus' chronology is a bit more complicated, so spoilers ahoy:

Sin was created in a war between Zanarkand and Beville. Sin was Zanarkand's ultimate weapon, and summoning it both saved and destroyed the civilization. Zanarkand lives on in a dream-like state, and the energy of the souls sacrificed gives Sin its power. Zanarkand won the war, but Sin was uncontrollable, destroying what little of Zanarkand was left, and moved on to terrorizing the populace of Spira.

A millennium passes in both Spira and Zanarkand. Jecht was a citizen of the dream version of Zanarkand, but is somehow transported to the real world of Spira. He is recruited by the summoner Braska, and along with Auron they go to fight Sin. The only known way to defeat Sin at this point is to use the "final summon" which kills the summoner and one of his guardians (Jecht volunteers as the sacrifice). The final summon defeats Sin, but is then possessed by the spirit behind Sin. Sin recuperates (the Calm) and then the cycle repeats.

Jecht uses what control he has left, and enlists Auron's help to bring his son Tidus out of Zanarkand. Their hope is that together with Yuna, the team can find a way to break the cycle and bring about the Eternal Calm.