Competition within a company for a candidate?

I applied to a few positions within a company. They’re all in different states specifically the two I’m interested in is NY and NM. I’m curious if recruiters can see that you’re applying to these positions, and if one team moves faster does it push the other team to reach out. (Workday System)

My situation specifically is I reached out to my potential manager on LinkedIn with my pitch and she said she’ll follow up with the hiring manager. The position I really want is in NY. But NM reached out first.

Just curious as I try to navigate committing to relocating in vastly different parts of the country. Also, should I mention my primary interest?