You need a few fairly cheap tools to perform the setup, but they are well worth the investment. Being able to setup your own guitar will save you lots of money in the future. If you've re-stringed correctly, I'm pretty certain it's your setup that needs help.

Oh, and I do know that different brands of strings have different tensions and can require a new setup, despite what others have said. Most, if not all string brands list the tension info on the packaged and websites- it's easy to lookup.

If your guitar was set up with low action before, and now these strings have slightly more tension, it could easily be just enough to pull your trem up and lower your action too much, which would cause a problem. A setup is the only way to fix this. Buy the tools and do it yourself, pay someone to do it, or put the same brand of strings you had on before. I'm pretty sure these are your only three options. I wish I could look at your guitar and fix it for you. I love this kind of work.

I asked precisely the same question a couple of months ago when i changed from Fender Super Bullets to Ernie Balls, same gauge. I had a bit of worry with the 'rattling', but my Strat soon became best mates with the Ernies.

Don't worry about it.