Like most phobias yours is totally without logic, it's what makes them such "fun" to have to deal with. Bit like my fish phobia
It simply is not possible to inject sufficient local anaesthetic solution into a vein, simply impossible, the veins in the area where the injection goes are so small and delicate they would be utterly destroyed within the first 2 or 3 attempts.
Remember too, the murderous dentist would be aiming for a structure about 1mm in diameter which is 3cm buried inside the tissues and completely invisible.
Injecting into an artery with a dental needle is simply impossible, the arterial wall is far too thick and the fine dental needle won't penetrate it.
If the solution contains adrenaline then the toxic dose is higher, 5mg/kg roughly. Even at excess of this dose, death isn't likely.