Check for spark when it dies, either use a timing light, or a spark check tool.
Yes. That's a very good way to see if you have a deficient coil or CD, or one of the low-voltage devices, such as the exciter or pulser. Likewise, it's possible that you have a float or choke that is stuck, allowing too much fuel. In that case, a cold motor wants a lot of gas, but as it warms, it floods. What does the condition of the plugs look like?
When you say you replaced the primer bulb, did you do the entire fuel primer hose, or just the bulb?
Assuming you find that the ignition fails after a minute or so, you have a marginal electronic component; in that case, grab your Factory service manual, and check the resistances of the exciter, pulser, CD, and coil.
Incidentally, never specify a Tohatsu OB by year; always model/serial.
