Lawrence is a bit dated, but what's old is new again!
The nicknames you chose freshen it up a bit too!
Go for it!
I think you should trust your gut, as your initial instincts seem good. It's a great name!
My husbands grandfather was Lawrence, but I've never thought of it as an old name. We had it on our list for a while before we found other names we liked. I think Larry is the dated part. Lawrence is classic and soft and handsome. If I were to use it, I'd use Ren or Law as nicknames so I think that works great too.
Most importantly, use the name you love no matter what anyone says! It's a handsome name and a handsome combo you've picked. ^_^
Came across a Lawrence in the supermarket today, a boy who looked about 8 or 9. It's definitely wearable on the right sort of boy, and he wore it well. I don't like nn Larry, it sounds very dated, and Laurie feels very feminine, though I do know a Laurie in his 40s. On a child though, I think it's too close to the popular Laura and Lauren :/ What about Lawro? Feels a bit more up to date and has the trendy -o ending. My primary association with the name though is my old maths teacher, who was a Mr Lawrence.
I love Lawrence and your choice of nicknames. Go for it! My only warning: if you really, really detest Larry, be careful - your son may get to be 14 or 15 and decide to go by that instead! Of course by then it may sound fresh to you. That being said, Lawrence is a classic name, not at all dated. I think Lawrence would fit in nicely among classmates like Louis, Henry, Charles, etc. but isn't likely to have several Lawrences to compete with. Good choice.
I'm strongly considering Laurence (I like the French spelling) for this baby, if it's a boy. It's an ancient name used for millennia and as such can never be trendy. The original Laurence has a great story attached to him if you're interested in such things. Larry is definitely a middle aged nn, but the others you mentioned are young (ESP the irresistible Laurie).