You can find some good older vehicles for that price range if you look hard enough. If the performance with the half-ton and 5.4 is barely getting by then I would personally skip it in a 3/4-ton unless it has 4.30 gearing behind it. Remember the 3/4-ton trucks weigh more, so unless gearing is much better power wise it will feel weaker, however brakes / suspension etc will be a lot stronger.
As what the others have said look for either an 8.1 (Chevy / GMC) or Ford V10. The V10 came in two different types, there is the 2v V10 and 3V V10 if you can find the 3V which I believe is 2006+ those come with Ford's 5-speed Torkshift transmission.
I went with an older Ford Excursion with the 2V v10 for the same reason's as you. I have a work truck, so mine sit's all the time and I paid around 9k out the door for it about two years ago now.
