My suggestion is that every bit of energy goes into Dragon 24/25 and Spider 25 basically for weeks or more.
Lots of folks even in my own clans just log in and spend a little bit just doing dailies and maybe working on leveling a bit of food. They may run 45 Dragon runs overnight sometimes, or even most of the time, and they wonder why they are being left behind.
You said you don't want to think about a go-second team, so realistically you have to run Dragon 20+ for weeks, with all your energy and maybe all your gems for energy, before you'll see speed improvements. No leveling food champs unless you're solo farming dragon. No going hard on some random Ice Golemn tourney, or running IG at all. Or anything else that won't help your speed problems.
And, if you don't think this is a process that will take a full month or more, you have to adjust your expectations. This will be a long process. Fortunately, people that already have a 340-360 Arbiter and a 320 high accuracy Warlord, they have to run exponentially more dungeons than you just to find a piece that improves their account even in the slightest, not even just with speed. So you can definitely close the gap, but only by being hyper focused.
That's my best advice at least.