Dead foot dead engine works fine as ONE means of identification, but is not of itself sufficient (see later comment about shutting down an engine). You could be experiencing some control asymmetry. Unlike some light twin training that you may have received, don't be tempted to give it a boot-full of rudder to keep straight, because if you do, it will roll badly and you will be in serious trouble. If you are doing the exercise in a simulator, it will probably crash to teach you not to do this. Just 'squeeze' the rudder enough (some describe it as 'step on the ball') to keep the ball within half a diameter of centred.
Obviously first job is fly it, second job is clean it up and third set max continuous thrust (many trainees miss this step in their haste to get on with the shut down drill). Whether or not you can engage autopilot for continued climb will depend on SOP and the actual autopilot fit, though obviously it is a better and safer operation with the autopilot offloading you during the next stage which is the identification and shut down. Some old B737 Jurassics don't take kindly to OEI autopilot operation, though if properly trimmed they should be OK. Ditto the simulator.
Then - and again unlike your light twin training, there is absolutely no rush to do this - go through the identification process at a safe height. In real life, jet engines rarely just go completely dead at exactly V1, even though we train a lot of this exercise to satisfy the licencing people. Same with engine fire - it is extremely rare to have fire with simultaneous total loss of thrust. So, in the real world (as opposed to simulator) chances are some thrust will still be produced and this would help you gain altitude and acceleration through clean-up. By 'sitting on your hands' for a minute or so you are using the dying engine to your advantage. But even if the exercise is a 'dead cut', no matter, no rush.
Before doing anything drastic like shutting down an engine, you should be identifying and confirming with your other crew which engine has failed, and indeed whether it has completely or only partly failed. EPR, N1, EGT and vibration indicators provide the clues. Don't be suckered into believing it is only a loss of oil because the oil pressure light is on or the gauge is reading low - that is usually a consequence of the engine running down. If it IS only loss of oil pressure, the engine will probably produce thrust for quite some time - more than enough to gently throttle it back and reach for the QRH.
Some SOPs actually require that an engine will only be shut down with TWO, or more, indications of failure. In the case of a fire warning with no loss of thrust or obvious over-temperature or vibration, that means the bell and the red light are counted as the two indications.