I looked at the data sheet. This lcd uses a communication interface that is different from any i2c display interface I've ever seen.
Which likely means you may not find an Arduino library that can control this LCD device.
The part to look at is page 8 (page 9 of the pdf)
That explains everything as to the i2c addressing and communication protocol.
That figure shows reads vs writes and each read or write uses 2 addresses.
To communicate with the LCD you actually must use both addresses
The lower address is set using LCD board pins 11 and 12)
pins 13 and 14 set I2C mode and must be set as indicated on page 5 of the datasheet.
To see how it all works look at the bits after the "S" bit.
For writes you see:
For reads you see :
Look at all 8 bits as whole and focus on the the right most bit which is bit 0.
You see all 8 bits of the Address that is put out on the i2c bus.
By convention slave addressing is only 7 bits with bit 0 being the Read/Write bit.
But that isn't why there are two addresses for this device.
Lets assume ID1 and ID0 are both 0 (which it looks like they were during the scans)
(BTW, offtopic but FYI, an I2C scanner should never scan below address 8, as 0-7 are reserved for special functions and can cause problems like a lockup on implementations that implement those special functions.
I wish all these i2c scanners would clean up their code to conform to the i2c spec)
So with those bits you end up with write address of 011100R0 and a read address of 011100R1
It is the RS bit that creates the 2nd address.
Here are the four 8 bit addresses:
01110010 - write with RS set to 1
01110000 - write with RS set to 0
01110011 - read with RS set to 1
01110001 - read with RS set to 0
But with the 7 bit addressing convention you end up with only 2 address as you only use the upper 7 bits of the 8 bits
0111001 - 0x39
0111000 - 0x38
With 7 bit addressing (which is what Arduino uses) bit 0 of the 8 bit address is implied by read vs write with write setting the bit to 0 and read setting the bit to 1
0111001 - 0x39 read/write from/to here will read or write with RS bit set
0111000 - 0x38 read/write from/to here will read or write with RS bit clear
So this is how you use the two addresses
0x39 - write to here to write to DDRAM/CGRAM
0x39 - read from here to read from DDRAM/CGRAM
0x38 - write to here to send a command
0x38 - read from here to read status & address counter
What this means that none of the "LiquidCrystal" type libraries I've seen will work with this device.
The hd44780 library includes a hd44780_I2Clcd i/o class but the way this LCD device works is different than the way the PCF2116/PCF2119x chips work.
This device uses a different address to set RS vs the other devices use a single address with a control byte to set RS.
This means that a new i/o class would be needed to work with this device.
It is a very small/simple i/o class as the code would be very similar to the hd44780_I2Clcd i/o class.
I can take a shot at putting one together but I REALLY don't like tossing out code that I cannot test and trying to debug things remotely is quite difficult.
Basically, it will either work out of the gate or I'll need to have one in my hands to test with.
I'll PM you with the code and we can report back here once it works or doesn't work.
In the mean time, please post some photos of your actual board front and back and your wiring so we can look it over.