I will smell another trimmer burned, if you use that schematic for 1A .....
Try this one, instead
(it works indifferently with LM317, LM1117 or LM338)
And remember that the current set resistor take the same current of the load, in all these types of schematics, so you need that it can hold the current and dissipate the related power, or it just burn .....
EDIT: about the different chips, is not matter of dissipated power, is matter of
internal dropout voltage ..... is not that LM1117 dissipate less than LM317 in the same circuit and with the same values, both of them need to dissipate the same power and amount of heat ..... is when you have a power supply voltage that is near the FV of your diode, that LM1117 gives you an advantage.
Trying to explain .....
LM317 have a typical internal dropout around 2,5V, where instead LM1117 have a typical internal dropout around 1,2V ..... if you are using, as example, a PHR diode, with a FV around 5V, with an LM317 you need, as minimum power supply voltage, the FV (5V), plus the dropout on the current sense resistor (1,25V), plus the dropout of the regulator itself (2,5V), plus at least half volt for safety, so 5+1,25+2,5+0,5=9,25V (can say 9V, but you are at limit), where instead with the LM1117, that drop 1,2V, you need 5+1,25+1,2+0,5=7,95V ..... it does not look too much difference, but with some setups, it can make the difference between a working and a non-working assembly
As matter of power dissipated, instead, if you have two identical circuits, both using a LD with 5V FV, and both powered from the same voltage, say, 12V, both the ICs need to dissipate the same power in heat, cause all the voltage that is not used from the load or dissipated from the sense resistor, is dissipated from the chip as heat ..... so it don't care that one drop 2,5V and the other 1,2V, internally, cause the difference, at the end is the same .....