Electronic – Mosfet switch not turning off

Your results are inconsistent with expected operation.
Either you are not doing what you think you are doing or the MOSFET is damaged or your test meter is of very low "ohms per volt".

Test1: Connect test meter with -ve probe to ground and +ve probe via 1 k to 3V3.
What is the voltage reading?

This should read 3V3 to a very close approximation.
If it does not give or throw the meter away and get a slightly better one :-).

Any meter that reads wrongly in that situation is a VERY poor one and useful only for eg battery testing.

Test2: Set meter to diode test range.
Measure Drain - Source.
With Source = +ve you should see a diode with Vf higher than a usual silicon diode.
With +ve on Drain you should see O/C.
With metetr connected either way G-D and G-S you should get open circuit.

Test 3 Ask Olin for advice.

Test 4: Check your circuit carefully.
Recheck MOSFET pinouts.

Try a new FET.

Note that MOSFETS are VERY prone to ESD damage - especially gate to D or S.
Handle with proper electrostatic precautions.

Report back.

^{simulate this circuit – Schematic created using CircuitLab}

Q1 and Q2 together are actually an SCR, but CircuitLab doesn't have an SCR symbol. This is essentially a crowbar circuit, with the input left unconnected.

F1 should be some sort of current limiting device. You said "stay off" but didn't specify for how long. A fuse will stay off for a long time. Replace this with a resistor and it will stay off until V1 decreases enough for the SCR to unlatch.