Couple things I'd have done differently:
1) Altitude is life. You gave the Spit a run for his money, especially considering he had a 5000ft altitude advantage on you, but ideally, I don't give an opponent an advantage of FIVE feet if I can avoid it. Don't fly right into the fight. Get yourself some altitude. This is ESPECIALLY important in the Corsair, which neither accelerates at level flight nor climbs particularly well. 15,000ft is generally good for the Mains. 10k should be adequate most of the time in the DA, unless you're hunting Tempest dweebs, who hover around 20k. Some of the larger maps in the Mains have more territory between bases so the fights get pushed higher, cruise altitudes of 20-25k However most cases, 15k is sufficient.
2) You played right into the Spit's hands. There's only six aircraft I'd want to use against a Spitfire Mk.I in a low-speed turning fight on the deck: Hurricane Mk.I or IIc, F4F, FM-2, or either Zero variant.
There's two key points of preparation in a given match-up. Knowing your plane is only half the equation. You need to know your opponent.
The F4U is significantly faster at all altitudes, anywhere from 50-100mph in the 1A depending on the altitude. Additionally, the F4U-1A will actually out-accelerate the Spit I at any speed, and above 10,000ft will out-climb her as well. You're far more rugged, and much more heavily armed. Finally, high-speed maneuverability in the F4Us is nearly second to none, with only a few aircraft that can match.
Where the Spit I has the advantage is pure turning ability at low speed, and climbing ability under 10,000ft.
It's REAL tempting to get your flaps out in the F4U and start turning, however it's easy to get sucked in to a bad situation that way, as what happened to you. Eventually you probably would have had the Spit. Unfortunately, he had friends to bail him out of trouble (including one you should never, EVER try to turn with: the Zeke that finally got you). By then you were low and slow and commited to a 3 vs 1 fight against two opponents that could handily out-turn you, and a F4U-1C with energy to spare. You probably could have gotten away from the Spit I, but not the others.
The F4U can fight in the weeds at low speeds and it's good to know how to handle her in that situation, but she's best between 250-400mph, especially with altitude to work with. You want to keep her faster and use the vertical. The F4U doesn't have a spectacular rate of climb, but in a close fight that's less important than her ability to zoom. The F4U is a heavy plane with a ton of horsepower, so with enough speed she can pop up like a rocket and catch a lot of opponents by surprise. She also winds it up very quickly in a dive. Additionally, because she's so sturdily built she can attain some incredibly high dive speeds without breaking up, AND maintains control authority.
Try to avoid constantly turning on one plane. Use high and low yo-yos with lag pursuit: In other words, don't try to lead your opponent through his turn to get inside him. Let yourself fall behind, and use the vertical either above or below the plane to get inside him instead.
What I would have done when I picked up the Spit in that situation, was to go shallow nose-low and continued straight, full power with WEP to build speed. I already know the Spit's going to try to dive on me. I'll let him. I may use a shallow turn in one direction to take me back to friendly territory to avoid the dogpile you ended up in. It also helps me keep an eye on him. Diving does two things: It gives me more speed to work with, and gives me more time because as our airspeeds increase it will take the Spit longer to catch me. Also, if the Spit isn't paying attention I can force an easy overshoot if he doesn't rip his wings off trying to pull up. This may be ALL I need, as I may be able to generate enough speed the Spit has no hope of catching me, leaving me free to grab altitude and engage from a favorable position.
Typically I'll wait for my opponent to close within 1500yds, then execute a low yo-yo INTO my opponent. I pull through the bottom of my maneuver and enter a zoom climb to gain altitude, and pick up my opponent IMMEDIATELY. If I've forced the overshoot, all I need to do is roll over and can drop right on top of him for a shot. From there it's just gunnery. If he was careful of his airspeed and manages to pull up in time, I'll wait for him to reset, then level out and go back into a shallow dive again to regenerate airspeed, and repeat the process. Eventually I'll either neutralize his E state or work myself into a position of advantage, or I'll generate an opportunity to reverse him and get a shot.
For more specific tips on the F4Us, check out the AHWiki. There's some differences due to variance in top speed, climb, acceleration, etc, but what works in one will work in the others.