Yeah, that's roughly how my center does it, definitely go ahead and get your application in if you can, I know my center keeps them on file for a while and even if you don't get picked for the next upcoming round of hiring you might potentially get an offer for 2 or 3 classes down the line. I think we usually do about 3 rounds of hiring a year, so it may be a little while before you hear back. There was one guy who took the test at the same time I did and ended up being in the next class, so it was probably about 6 months between him putting his application in and when he started.
When I started (about 8 years ago now) I went to a hiring event where we took the aptitude tests and got an interview on the spot. After that, we scheduled a job shadow to come in for an hour or two listening to calls being answered dispatched to see what the job is actually like.
To give you an idea of the timeline, I think the hiring event was around August, I got my conditional offer in September, and class started late October.
After getting the offer, I had to do a hearing/vision/drug test, and sort of a psych eval with the county psychologist. Some other agencies have more requirements, I've heard a lot of places have you do a polygraph test.
In addition to 911, in a similar vein there might be some similar dispatch jobs you can look into as well. Some police/fire/EMS departments s might have their own dispatcher in-house, some agencies like state police, park rangers, game wardens, animal control, etc. might have their own dispatch separate from 911 as well. Public transit and railroad companies often have their own police, so do some bigger colleges.
Or if you're not bothered by the ethics, some bigger security/private investigation companies like the pinkertons have their own sort of dispatch roles.
And if there's no residency requirements and you don't mind the commute, you can look into your surrounding cities/counties to see if they're hiring. We have a couple people who work at my center who actually live pretty far out of the county.