The TL note said it's not a romantic partner. My guess why he is so timid is because he is just glad to have someone who isn't weirded out by him and she is flustered because he seemed less tense or something along that line.
Anyways, why the fk is he running from the police when he did more than good? Sure a lot of annoying comments from the cops the past few times but he never committed crimes as far as the story told so far.
More wondering how someone in their mid-20s is "an oldie", tbh
But to your second point - it's well-established that he's had to deal with a
lot of shit simply due to who his parents were and what they did. He had zero hand in it, and he's been ostracized by classmates and coworkers and employers and even landlords ever since.
Him running from the cops in this moment is entirely on-theme and consistent with his character. It's not just "annoying comments", it's a constant deluge of gossip and rumormongering that have resulted in him being evicted, fired, or pushed out purely on the 'social image/stigma' of what befell his family when he was a hapless child. That
will wear anyone down over time, and he's been economically vulnerable and on the edge of poverty his entire adolescence and young adult life as a direct result of those "annoying comments".
Plus, dealing with the cops in this situation would just more hassle for him, as
she's also there and would get roped in. What if the cops found out who he was, and then found out he's rooming with a 15-year-old high school student? The potential for tons of new headaches (at minimum) are rife, and so high-tailing it was, arguably, the correct choice on his part.