I researched this oil leak issue a few years ago after a 1996 Contour 2.5L 24V DOHC I have started leaking from the crank seals when I switched it to Mobil 1 for the first time at 116K miles when I bought it from the company I worked for. This is about the only scenario where there can be a leak "caused" by synthetic oil: If the engine temp has spiked high, very high, during a big issue like a hose or water pump failure (which my Contour had), the dynamic seals can weaken somewhat and take a bit of a thermal "set" by hardening. The dino oil will tend to form a varnish that will basically fill in the resulting weak seal diameter. When you switch to synthetic, it's superior detergent properties will clean away the varnish deposits and then, due to the improved lubricity, it can form a small leak.
I switched back to the 5W-30 semi-synthetic Motorcraft oil and after about 20K more miles the leak eventually stopped. It has 196K on it now.
It doesn't happen often, but it can happen. It's the only time I've seen it happen. I've been running synthetic in everything I own since about 1990 and this is the only vehicle I ever had an issue with. I would definitely give it a try, the overwhelming odds are you won't have any issue.