Don't let some of the miserable people here ruin your goal. If you live driving distance to base and understand the current situation here, then by all means do what you can to get hired. Your TT is low, but the quality of it sounds pretty good. Attending job fairs is a big one, especially with your TT, and meeting recruiters face to face can boost your resume to the top of the pile. Don't expect a call from just submitting an application and sitting by the phone.
Do what you can to get hired at the big 3, keep the apps up to date, but for the meantime this place is much better than being at a regional that you need to commute to, and if the music all stops tomorrow, I'd much rather be stuck here. I have my apps up to date with the big 3, but being here and living in base vs commuting to a regional isn't even a comparison.
Yes, comparing ourselves to the bigger guys we are very underpaid, and we're all fighting very hard to change that. That said, the people who suggest staying at a regional, or leaving for a regional are delusional. I'd need to be a 10yr Captain at my last regional to make the same hourly rate as a 2nd yr FO here.
Good luck, and if/when you get here, remember. You are a pilot. Not a cleaner. The only time you should be wearing gloves is when you're heading out in the cold to do a walk-around.