I had a couple additional thoughts which might be of help to you or anyone else interested in the subject of your thread.

First, on the question of shooting line versus just casting longer using a double haul, I know that I loose some accuracy when shooting line. I much prefer to make a longer cast holding more line in the air while double hauling. Others may have different experiences they would like to share.

Second, I was going to suggest underlining your rod which may help you increase the line speed - before you mentioned that you are using a 3 wt rod. For longer casting, either just double hauling or shooting line, my preferred rig is a TFO BVK 9' for 5wt underlined with a WF4F line. Some people on the BB in the past have suggested that the BVK is too fast and overline it, but I doubt that helps them make longer casts.

The 3 wt will shoot some line, with or without a double haul. But it won't be the easiest rod to learn with, especially if it is on the slow side.

I did run across a couple Orvis videos on YouTube this morning, one on long distance casting shooting line without a double haul and one on double hauling. I wish I had seen these way back when when I was struggling to develop a decent double haul - being a visual type, these videos certainly would have gotten me where I wanted to be more quickly. Both videos are fairly short and are well worth the nominal investment of time.

One last thought. For a year or so, I had a spey rod - 14' for 8/9 ( spey line ). Just learning stuff about spey casting, especially investing a good amount of time in Simone Gawesworth's book and videos really helped my understanding and the practice of casting even single hand rods. The spey rod didn't work for the fishing I was doing so I gave it up - but the work I put into it was a real benefit and worth the effort. Simone's videos on YouTube are usually fairly long, but he one of the best in world and quite the entertaining personality on top of it.

John