What is the ruling on accidentally touching things that are wet with urine, and what should I do about my feeling that the house is najis (impure)?

Ignore the whispers (waswas), and do not pay attention to them.

If you advise your father with gentleness and kindness, or if you entrust your mother to advise him not to sit on the seats with his impure clothes, then this is good. Then know that something is not judged to be impure merely by doubt, for the original state is purity. If the impurity is dry, it does not transfer. So, cast away doubt, do not pay attention to it, and do not judge that impurity has transferred except with absolute certainty.

If you are certain that something has become impure, then its purification is by pouring water over it until it covers the location of that impurity. You are not obligated to purify those things unless you touch them during prayer. If it is difficult for you to remove those impurities, then all you have to do is pray in a place that you do not know to have been affected by impurity. You can also designate specific clothes for prayer so that you avoid being afflicted by waswas regarding whether the clothes you wear while sitting at home have become impure or not.