Hey hannaugh,
I actually use both hands to block unwanted strings, but it can take a while to get used to this. If I'm working on a new lead solo, I break it into sections to figure out how I need to block the strings. I think you're trying to do it all at once, and that's when it can get messy as you say.

Break it up into short licks, and figure out whether to palm mute with the RH, or whether to use the LH to block the strings. The most important thing is to slow it down, and play it the way you want it. If you hear something you don't want in there, figure out how to stop it. After you play it the way you want it slowly, then you can build the speed, because you know what you need to do and when to do it.

I hope this helps you, and let me know how it goes.