I've yet to more than the first episode of season 3. Don't remember too much growth in seasons 1 and 2, beyond going from not visiting Jason in the hospital while sleeping with Jason's girlfriend to accompanying him to Mexico and talking him out of a risky procedure. He was still an unreliable teammate (drinking before practices; taking depressed Matt with him no less) whose relationships, according to Lya, lasted 20 minutes.
Well, I'll grant you that most of his growth took place in Season 3 (watch beyond the first episode, believe me!). However, off the top of my head, at the beginning of Season 1, he was hungover, banging Tyra and random rally girls, cruelly dumping Tyra (or rather, indifferently allowing Tyra to "break up" with him whilst teeing beer cans off of a cliff), and ignoring his best friend's paralysis because of his discomfort, guilt and depression. By the end of Season 1, he'd started to turn things around, reconciling with Jason; proactively defending Lyla to Jason; convincing Lyla to follow her heart and go to the cheerleading competition; and trying to make amends with Tyra. During Season 2, we saw him progress further, accompanying Jason to Mexico and preventing him from making the worst mistake he could make; falling in love with, and pursuing, Lyla; and focusing on that goal single-handedly, which brought out lots of other charming (and more mature?) aspects of his personality, e.g., no longer sleeping around; drinking less; going to church (!); and committing to an actual non-football activity (the radio show). But Season 3 is really where he is transformed -- it is Season 3 that follows Tim supporting his best friend unconditionally, helping Jason achieve his dream (in one of the best episodes of the entire show!), making money as a result of (sort of) hard work, turning it around on the football field and mentoring a new quarterback, and, perhaps most showcasing his growth and progression as a character, making a relationship work (and work beautifully) with Lyla Garrity. Through his relationship with Lyla, he really became a man; he embraced the role of her protector and caretaker, and he stepped up big time.
Anyway, despite all of this progression, of course Tim wasn't entirely devoid of what made him TimRiggins in the first place, i.e., the drinking, partying, and dislike of school, organized football practices, and any kind of work. Those personality traits follow him not only through Seasons 1 through 3, but through Season 4 as well.
Also think in seaons 1 or 2 he definitely would have ended up in a 20 minute sexual relationship with Becky.
Possibly in Season 1; and possibly the beginning of Season 2; probably not end by the end of Season 2 and certainly not at any point during Season 3. That said, even Season 1 Tim wasn't scummy. He did get into that relationship with the neighbor lady, and he certainly treated her son well. So it's certainly possible that Season 1 Tim would have reacted the exact same way with Becky/ her mom given the circumstances. I can't imagine that Season 4 Tim wouldn't have slept with Becky had he not banged her mother already and were he not living in the trailer. What's the expression: don't shit where you eat? Or something like that.