operational amplifier - Op Amp as a simple switch

I want to build a little voltage switch that switches on a buzzer and a relay when a certain voltage is reached. Please have a look at the schematic.

I use a trim potentiometer to set up the switch voltage so that it equals the zener voltage, in this case 3.3 volts. Then the Op Amp (741) switched to Vcc and drives the transistor (PN2222A) which switches on the 12V Relay and the buzzer. The Basis resistor is set in a way that I will have a Ic of around 150 mA.

So far so good.

My problem is that the Op Amp is not "ideal" and does not switch to 0V when the Voltage on the positive input is lower than the Zener voltage. There is about +2V, which is obviously enough to drive the transistor, so the relay/buzzer is always on.

There is also not a "hard switch" from 0 to Vcc, the Op Amp slowly increases the output voltage and then suddenly goes to Vcc.

I am aware of the DC offset problem, but how can I compensate it if I only have Vcc and 0V?

One other solution I found on the web was to put a feedback loop in. How do I dimension the feedback resistor?

These might be very basic questions, but I would appreciate your help.

Thanks in advance.