The previous administration is still running the town, so how is Braxton going to arrest them when they control the police force?

He became mayor because he was the only one to qualify for election in 2020, because the incumbent declined to submit paperwork (because they weren't planning to actually hold an election; the last proper mayoral election was at least 60 years ago, according to the article). The city council responded by holding an unannounced (and probably illegal) special election to re-elect the previous mayor, which no one knew about so Braxton couldn't apply for it. So the previous mayor has simply continued doing the job, with the backing of the council and all his officials. That's why Braxton is bringing a lawsuit against them.