An LM350, can I have both constant current, and specific voltage?
In constant current mode, it will output whatever voltage is necessary to push (in your case) 1.2A through the load.
This will be limited by the supply voltage (minus a bit for the regulator drop) as obviously it can't output a higher voltage than goes in. So you can't expect it to put 1.2A through an e.g. \$ 1 M\Omega \$ resistor unless you have a supply of 1.2 Megavolts handy :-)

Foe example, if you have a 1 ohm resistor as the load, the voltage at the load will be:
 1.2V (\$1.2V \div1 \Omega\$ = 1.2A)
 If you have a 5 ohm resistor as the load, the voltage at the top of the load will be 6V (\$ 6V\div5\Omega\$ = 1.2A)

You can't set it up for constant current and constant voltage, as to keep one constant requires varying the other.
However, with a static load in constant current mode you can set it to drop 2V - e.g. \$2V\div 1.2A = 1.6666\Omega \$ resistor needed. So if you have a load of \$ 1.6666 \Omega \$ on the output of a constant current of 1.2A, the voltage will be 2V.