Sounds about right. But you'll find more long-term happiness if you choose batteries capable of at least twice what you actually expect you'll need. The lower the internal resistance of the battery the better.
A battery with twice the mAh rating is no better than a lower rated cell if it...
There may be some here who can answer your questions but you may have to wait a while. WL was (is) such a ripoff they don't get a lot of respect around here. That being said, I wouldn't mind having one of their original 473nm Spyder lasers.
Power=volts x amps
If you double the battery voltage, a good driver will cut the current draw by half.
If your laser will achieve full power with a good 18650 I'd go with that and use a protected cell. The 18350 types are usually unprotected, meaning you can drain them enough to ruin them...
OK, first things first. If you're set on starting with 1 watt+ then before you ever order the laser ORDER SAFETY GOGGLES! 1.2 or 1.6 watts is plenty to burn your house down accidentally. That's enough power to screw up your eyes.....for life.......in an instant of inattention.
If it was running at 4+ amps you wouldn't be able to hold it in place with your finger like that. A heat sink that small would heat up to boiling in 30 seconds with that diode.
If you have a DMM check the voltage at the diode pins. Your power supply voltage may be getting eaten by lead...
That just sounds like too much current. Too low a voltage won't damage the diodes. Try an LM317 driver with an adjustment pot and turn it up slowly. And build a test load for dialing in your driver. Search this forum if you want instructions on that.