Other considerations
I agree that you need to be contacting agencies now or soon, and initiating the process. Check their websites for contact information, FAQs, etc. You have a good chance of having some dead time, regardless, and should not count on being able to get out of the Corps and start immediately, or even quickly, with an agency.
Consider being able to use that dead time to do a semester or quarter, as the case may be, at a local college. I did my first 2 years of undergrad at a community college - best value out there. Darned cheap, and broadening. You'll meet a lot of people that you will not meet in the Corps or a 4 year elite university (such as where I finished by BS Ed.) Single moms, retirees, messed up kids, folks changing careers because their employer went TU after 15 years. The variety will be good for your people skills, which will be a critical part of transitioning to LE from the military. The mission and population are vary different, and I have seen people and agencies which have had serious problems with this transition.
Other thoughts: get a feel for how the agency treats its people. Watch for shift rotation. If it is done more than once a year after the training car, go somewhere else. Forced shift rotation sucks; it's hell on the body and mind, and morale suffers. If you can bid as needed, that's ok. 4-10s, for my money, are the best shifts; 5-8s cut your personal time, and the 12 hour schedules are real butt kickers. (YMMV; I'm old.) Be aware of the benefits. A take home car is not taxable for LE, at least under federal law, and means you don't need to have a car to go to work. It's probably worth most of $10K/year, at least $5K. See how good the family health coverage is; it is not hard to spend $500/month out of your pocket if you have kids. From what I hear, it could be more (I don't have kids, and the dogs don't get coverage).:biglaugh: $1K a month more in income is easily eaten by taxes and health coverage; pay alone is not the end of the calculation
Look for tuition assistance, and other similar programs. Be wary of agencies with rigid uniform rules based on appearance - mandatory hats are a real gripe to me; hats should generally be prohibited except for foul weather and helmets. A command officer who can't tell the difference between a parade ground and police work is a real menace.
Good luck with your endeavors.