Not much help but just reminded me of a car I had that did this.

Was a mk1 golf gti I bought when I was 18 (against my father's wishes) it was a really straight clean car and I was shitting it on the drive home thinking he's gonna go mad but also thinking it's so nice and a bargain, how can he! I basically spunked all my student loan and bursary on the car and insurance, hence him going mad.

I pulled up at home, turned the key off... Still running! Fook. Mass pannick to follow. Then followed lots of random wiggling of wiring, popped the bonnet and happened to disconnect an aftermarket fused live direct to the battery and she cut out. The thing is if I left this live disconnected all the time the car would turn off fine but when running the indicators would all be lit up solid and the lights on the dash too.

I never did get to the bottom of it and ran it like that the whole time. Sometimes it'd be fine, then others it'd just run on.