When Chase and Tate are fighting on the landing, in a wide shot, you can see that Harris Dickinson slipped out of and nearly lost his left shoe. It cuts to a different perspective and the shoe is back on.
Kya circles July 4, 1962 on her calendar, which shows it falling on a Saturday. That date actually fell on a Wednesday.
When Kya and Chase travel to Asheville, they arrive at a place that has a large "Hotel" sign in front of it. This is not the correct term for this kind of establishment. These places are called "Motels" not "Hotels". The room Kya and Chase entered was facing the parking lot which defines the motel concept. Hotels connect their rooms via a lobby and hallways.
The calendar in Kya's house should show 1963 and not 1962.
All the of the addresses of publishers Tate gives to Kya have ZIP codes. He gave her the list in 1962; the first ZIP codes were established on July 1, 1963 and were not in common use until the late 1960s/early 1970s.
Regarding the 50-star flag, Alaska and Hawaii were both admitted to the union in 1959. Kya's calendar shows the year 1962 when she was waiting for Tate to return on the Fourth of July.
One of the police boats in the swamp chase has low-profile, high-visibility blue LED lights. These would not have existed in the 1950s. Also, police didn't use blue lights at this time. Police would have used red "gumball" lights.
The US flag shown in Jumping's store is a 50-star flag. At the time of the scene (1953) Alaska and Hawaii have yet to join the union. The correct flag would only have 48 stars. The flag is also hung backwards, with the stars on the left instead of the right, as proper when a flag is hung vertically. This has been corrected in a comment below.
Asheville is approximately 6 hours from the coast of North Carolina. No one would travel that far to shop. Raleigh is half that distance.