Yes. All serial wires that go to one display need to go to the other. This is for redundancy, so that you don't lose any functionality should either of your displays fail.
If you're NEVER seeing ADS-B FULL, then you're either not getting a traffic portrait traffic from the ground stations, or there's something still not quite configured right. On the screen that doesn't have the transpnoder connected and configured, this would be expected. ADS-B partial is inherently not showing you all of your traffic. In this case, if you ARE getting TIS, then you're getting an pretty robust traffic picture, as you're seeing what the controllers do (this is equivalent, if not more authoriative, to ADS-B).
Here's an excerpt from our pilot's user guide that explains the priorities of different traffic devices:
Traffic Receiver Priority
Only one device can provide traffic information to SkyView at any moment. If you have more than one device that is capable of providing traffic, these traffic sources are priority ordered based on the completeness of the traffic portrait they provide:
1. SV-ADSB-470 with FULL traffic service: Full traffic means that you have an ADS-B ground station reporting, and that ground station is receiving radar traffic. This is only possible if you are equipped with an ADS-B Out device such as a SV-XPNDR-26X or Garmin GTX330 ES transponder. In this case the ADS-B system delivers all known ADS-B equipped traffic and all radar targets that FAA radar system can see.
2. TIS transponder from SV-XPNDR-26X / Garmin GTX 330 (when in an active TIS coverage area).
3. Flarm device.
4. Zaon device.
5. SV-ADSB-470 with PARTIAL traffic service: SkyView will annunciate “Partial” traffic when it has ADS-B reception capability, but that ADS-B reception does not include either ADS-B ground station coverage or radar targets included within those ground-based ADS-B traffic reports. Therefore, the ADS-B ground station is not able to convey a full traffic picture and cannot make you aware of all possible detectable traffic.
I suggest you get in touch with our technical support staff directly at 425-402-0433 so that they can help you with your particular configuration.