is it important to put antifreeze in our travel trailer over the winter?

Simply draining them and pipes could still contain enough water and that may be enough to freeze and split and yes, it will get cold enough there to happen. Might be able to push some compressed air through the lines to blow them out.

Yes you have to put antifreeze in it. You can't get ALL of the water out of the system and the repairs in the spring will exceed the cost of the antifreeze by a lot. Also don't forget to do the sewage system as well. That will cost even more if you break those parts. Hope this helps.

I thought antifreeze should always be added to the water whether for hot or cold weather to keep the vehicle temperature balanced.

You'll sleep better if you drain it best you can then dump RV antifreeze on all the sinks toilet and shower stall drain.