I strongly suggest that you have a local tree person (certified arborist) come out and give you an opinion on this one. You can certainly trim your tree now; it may bleed some, but you can remove any limbs endangering your home, especially since spring storm season is nearly upon us. But I suggest having someone just come look at it. They don't usually charge for that, or if they do, it's a minimal fee. Get their advice on the safety of the tree--if it's worth saving, or if any trimming will make it unsturdy/lopsided/more likely to split. I wish I lived closer--I'd make a house call for you!
You won't regret getting a professional opinion. As long as they're on your property, you might as well ask them about all of your trees. Get your money's worth! You can still do all the trimming yourself. Do this sooner, not later, so you can have an impact on this season's growth.
Be sure to ask the arborist if they recommend any special mulching, feeding, or watering process after pruning (for this summer's growth season). Happy gardening!