I really think your best option is to call. Give them a date and destination and then tell them you are flexible on dates and have them check several days before and after the dates you provide. Keep trying until you get something you like. Note it may take several phone calls to piece together an itinerary.

Don't be shy and hang up (politely) if you get an unhelpful agent. If you are trying to avoid the phone fee then write down the flights the agent found for you and then try to replicate the same itinerary on the website.

Keep in mind that I don't think its currently possible to mix UA and US on the same itinerary when booking online. Also make sure when you are searching online that you notice you have to go to a separate area to search for US flights (its not automatically searching both UA and US at the same time).

Hope that helps!