Your tires could be a problem. 35s on 18” rims is a lot of sidewall to flex. I’m going to guess that at 35” that the truck is lifted and they are AT or MT tires? Neither ideal for towing but shouldn’t necessarily cause sway. If they did it would likely feel loose at all speeds. I tow with a lifted 2500 truck with 35” Toyo ATs without problems.
I would suspect the issue to be more with the trailer. What are you hauling in the garage and how is it loaded? A vehicle? Is it loaded front end first or back? Are you loading the heaviest parts over or near the trailer axles?