Yes, you can control which messages are sent to each of the two synths by assigning a different MIDI channel number to each of them.
On the TT-303, switch the Mode selector switch to “MIDI”. The default MIDI In channel number is 1, but you can change this in the range from 1 to 10 if you need to as described in the manual for the TT-303.
On the xoxbox, select a MIDI channel number (maybe 2) that is different from the number you chose for the TT-303. See the instructions for the xoxbox for how to set it up to receive MIDI.
Now if you send MIDI notes from the OT on channel 1, they will only act on the TT-303. Similarly, if you send MIDI notes from the OT on channel 2, they will only act on the xoxbox.
The way to do that is described in the OT manual section called “The MIDI sequencer”. To try this out, first try using a pattern that has no trigs on the OT’s audio tracks. Select MIDI mode, and you now have 8 MIDI sequencer tracks. Select one track. On the Note Setup page, select the MIDI channel number for the synth you want to sequence and press the knob or press [ENTER/YES] to confirm the selection. Now you can enter note trigs on the MIDI tracks just like you would on the audio tracks.
If everything is OK, the OT should send MIDI note messages to the synth.
Try it out and let us know what happens.