If the recruiter who emailed you sent their name, then I would give a call and let them know that you have a pending move. As with all job applicants it’s hard to get through the phone layers to talk to someone, but if you have a name it’s sometimes easier.
This part isn’t the hard part. Once you get your offer letter, the PQ process starts. If you read back some of the posts here you’ll see that it doesn’t always go smooth. My advice is to set up an exam as soon as you get the packet. Same for the Dental. Many people send in their exams only to have the medical people flag for more tests or to order dental work done. While seldom serious , the process can take a long time and if you wait too late to start, your deployment date can come and go without you being finished. The bottom line is that if you aren’t PQ’d you won’t be coming down.
Keep us informed.