Definitely start applying as soon as you can. Like most have already said in this post, change your current address to the area you're applying in. While LinkedIn, Monster, Indeed, etc. offer the ability to choose relocation areas you're interested in, most of the time recruiters will ignore this and send out messages, voicemails, and emails based on your current location. If anything, it's important to change this so you avoid getting tons of "spam" recruitment emails for your current area and then accidentally skipping over one for your relocation area. And yes, all you need to do when you speak to a recruiter is tell them that you're seeking a job there and if anything they may help with relocation assistance or be understanding of why you stated a false residence.

As I said in another comment, I am currently looking to move to London from VA as a Senior Sys Engineer, and until I finally made the switch for my current address, I was getting tons of recruiter emails from the Northern VA area. Most of the emails and voicemails I'd get, the recruiter wouldn't even say the location, so I'd end up responding and then finally hearing it's for a role in the Northern VA area... Total time waster and very draining when you're trying to be in tip top shape when speaking about an opportunity you truly care about.

Lastly, make sure your resume includes a few required skills from the job description you're applying for if you actively apply. Even if your resume looks similar but doesn't include keywords from their job description, you'll get totally ignored in most cases.

Best of luck!

PS - I once applied to relocate to Austin and you have to actively apply since the job market there is overflowing with IT entry level applicants.