Engine stumbles, hesitates, at low speeds

You will have to have your vehicle scanned to retrieve trouble codes. with the codes diagnostics of the problem can be more reliable. It could be something as simple as a vacuum line cracked or off. check links below for more info, press the helpful button.
Problem 4 Engine Hesitates Stumbles Lacks Normal Power

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sounds like engine need some cuddle :) I mean, first thing to do is to check engine compression ratio. If that is ok, then make sure carb adjustments are correct and all fuel lines ar clean. Good luck!

check the manifold is not leaking on the intake side. also check and clean the intake top. carb clean helps to clean out all the muck out on to a cloth. also carb cleaner will help if you have a leak spray round the intake when engine is a temp. engine will cut out or speed up depending on carb cleaner used. were there is a leak.

Sounds like you have a misfire, that causes the ck engioje light to flash.

check alternator,check battery connection,fuel filter,air filter,check coils and module,spark plugs,last thing crankshaft sensor.

is the check engine light on? have you ever serviced the injectors? if not try that first because there are many things that give the symptoms you spoke of. but this is the cheapest.
vbninja E.A.P

MOST TIME CAR STUMBLE OR HESITATION ON ACCELERATION YOU.HAVE A SLIGHT LEAN FUEL MIXTURE.CAUSED BY STOPPED UP FUEL INJECTORS OR FUEL FILTER SLIGHTLY PLUGGED.

first make sure you are up on your maintainence.. plugs, wires, fuel filter, etc.. sounds like a fuel issue I would recommend running a fuel pressure test first..

Did you check earth connection of electric system?

it could be TPS or MAP. corolla dont call it a map but its on the firewall near the brake booster check your vacume lines see if any have come off or cracked or dry rotted. vacume leak could be the problem. there are a lot of them to check.