How bad would my car run without?

yep, it probably would be worth it

Since you already have the O2 sensor out, I'd replace it as well. You should be able to replace the cat for about $300 including labor if you shop around. Those T-Birds are really nice cars. Does that have the 5.0 or 4.6 or the V6??? I have experience with the ford 5.0 and 5.8 and I personally think they are great motors. I'd spend the money to fix it, but it's up to you. FYI you won't get much for it if you try to sell it or trade it off if it's running like that. Everyone is right about the O2 sensor, it NEEDS to be in.

You cant even replace the converters for less than 500 bucks let alone fix anything else. If your budget is really only 500 bucks for a 12 year old antique car, I'd donate it to the blind or something and use it as a tax write off.

Just get a universal cat from Jegs.com or off ebay, cheap and a shop will weld it on pretty cheap, you should put your 02 sensor back in if its still out, it sends a signal to your Computer telling it what the Air Fuel ratio should be, So pop it back in get a Uni Cat and get it welded on and your good to go.

shop around, you can get the converter replaced for less than that, and why wouldnt you even if you spend the 500, you could turn around and sell it for much more than the cost of repairs.

The air filter out does in simple terms that...filters the airborne dirt and dirt and airborne dirt and dirt out of the air that the engine makes use of! Any airborne dirt and dirt drawn into the engine reasons greater friction, and placed on! placed it lower back on with the intention to no longer wreck the engine!!! a reliable sparkling filter out could make little or no distinction interior the quantity of air flow going into the engine...as to more suitable overall performance...it relatively is perhaps in simple terms your thinking, except the previous filter out grow to be very grimy and plugged!

disconecting your sensors is not recomended a the cars computerm relies on for information.timing etc. if you are happy with your car and it is safe, try to get a few estimates on its repair. not every one can afford a newer car.