I think its not important how much voltage Input ges into the LM317 it can be used as a Voltage Regulator or a current regulator (CC out) 7,2 V would be best input for a red diode so you need to drop your Voltage for your AC/DC adaptor i would use a resistor when its Output is CV when not use a...
How much mA do you need ? If 300mA you will need a 4,3 Ω resistor max. mA for a LPC-826 was 440mA i recommend using 350mA but you wont get that out of the adaptor...
I was using a aixiz module for the first Time all my DiY lasers was made out of other modules and I don't care about waste or not i use it 4-6 hours a day and i had the TEC laying around so i used the TEC