CAUTION - this is probably more information than you want, but when you get me started on this topic... ;D
If you are wanting to dress your 350 up like a '63 327, the biggest obstacle will be the crank case breather system. I think all the 350's utilized valve covers with the holes in them for the breather and oil fill. The 327 used a hole in the rear web of the block, near the distributer for the vent (not sure in 63, may have even just been a draft tube). Also the oil fill tube from a 63 327 would be on the front of the intake manifold. After I did this, I heard of of someone getting much the same thing done with a hole in the intake manifold. Perhaps a more prudent choice.
When I started my restoration, I was going to try all this on my 350 based 383 that I had in the car already (put the 383 in there back in the late 80's). But in the end, I started wanting to keep it more stock looking than I could achieve with the 350 block, and had a 327 built, bored and stroked to a 383.
This is how a 63 block would look. On the right (front) of the block you can see where the oil fill tube goes. On the left (rear) you can see the vent tube hole, just above the distributor hole.
Your 350 block would be like this (sorry, the orientation of the photo is flipped from the one above). You can see on the left (front), no oil fill tube provision. ON the right(rear) only the distributor hole, no vent tube hole.
My old 360 based 383 with Quadrajet carb. You can see the vent hole and the oil fill cap on the valve covers
I did start the process of making it look like a 327 by drilling the block for a tube that would give me a fitting in the location of the vent hole on the rear of the block
Adding a '64 intake manifold and Carter AFB let me put on the oil fill tube. So at this point the most obvious give away was the larger stamp pad that the 350 had. But I did know the engine would need a rebuild, so this was just tinkering to see if it would work.
But like I said, in the end I went with an actual 327 block. Since the 383 stroker cranks are large journal for 350 blocks, the engine builder put on a set of 350 main caps and had the mains line bored to match. So now the small journal 327 block could take the large journal 383 stroker crank
Some roller hydraulic lifters, so no worries about cam break in.
And the finished engine. I think this was before Vern Frantz gave me a few tips on a few details to get right. But a "327-300" with 370 hp and 420 torque. I love the easy power that this torquey motor has. I run the stock exhaust system with resonators. Has a slight rumble at idle and when you get on the throttle, a little roar as it gets up in the RPMs. Just enough to make a good sound track, but not loud enough to be annoying.