Black was trailing for most of the game. How can black not fall behind?

You were not losing all the way, but you did get beaten in the fighting a few times, so it is a bit hard to give general advice. In positions where you are locally strong, try harder to find severe moves, to make full use of the strength of your stones. But in positions where you are weak, be more careful

The lower right corner was a difficult fight, where you came out a bit behind. To mention just one or two alternatives, consider at S4. This separates W more severely, without provoking the corner to live. Or consider at T2, giving W two choices, both better for B than the game result.

After this, though, you played well in the center. When cut at L9, I think you were doing fine. It looks like you have good prospects of making territory while attacking the W dragon.

was bad shape. It may look strong, but just extending to M12 is much stronger. Liberties are important, and you want to keep up the pressure on the W center group. After at R11, your position was difficult again.

The submissive exchange S10-S11 should be very painful, to be played only if absolutely necessary to live, and maybe not even then. Consider S11 or Q11 instead. Either of these moves should give a better result than the game.

Both sides misplayed the right side life & death problem after B was enclosed. at T9 is a tesuji connection, winning the fight. When W misses it, B can play there to live unconditionally. Eventually though you got a ko and handled it well. When you won the ko and W got very little for it (Q13 should be at Q14), I think you were actually ahead.

Let's spend a little time looking at the position at move 122. B has enormous center thickness. How to use it? The wrong answer is to make it directly into territory, but that is what you did in the next few moves. This earned you only a very small territory directly in front of the wall, while giving W easy territory along the top and corner. By the time W played C14, he was back in the game, probably even winning. Things went downhill from there.

A much better strategy is to use your massive thickness to attack, and to count on making territory elsewhere. You have enough strength to support an invasion at F17 or G17 immediately. Or you can prepare with L17, which is large in itself and makes the invasions really threatening. I suspect a strong AI program would tell you to invade at C17 and take the corner territory, since the thickness W gets in return will not be worth much. All of these options make good use of your center thickness.