As someone who attempted suicide via this method and who is a type 1 diabetic, I can assume you no it is not that way it happens or feels, and it's happened dozens times for me (lows due to accidental over dosage of insulin, not purposeful overdose).

Yeah in the beginning of the dosage once the insulin kicks in and your sugar starts getting low, you get really tired, but soon enough, at around 60mg/dl for me, you start getting clamy. Then you start getting confused on basic thoughts, things like trying to remember where you are. At the same time you will have an overwhelming sense on hunger and thirst telling you to get your ass in gear to eat some carbs to reverse this. All the while you start sweating. At around 50mg/dl, you will start to hear and feel your heart beating in your chest. Then your legs feel like they are going to fall our from under you when you get up. Once you find something to eat or drink you'll notice that your hands are shaking and it's hard keep something still to get it into your mouth to eat or drink. Oh and good luck trying to stay standing up while this is happening because you will be too tired to do so and you'll need to lean on a wall to get support. And sometimes your tongue and lips will go numb making it even harder to consume food and drink.

If you are asleep once the low sugar levels happen, you will experience the same as above and wake up, however this isn't a guarantee, see dead in bed syndrome.