I've owned and played several of both, as well as Gibson SGs, other lesser-known Fenders, and my favorite, a Gibson ES-335. In the early 80s I was in a cover band and needed to duplicate the sounds of both Strats (Disco/Funk) and Gibson Humbuckers... a friend of mine turned me on to the new (at that time) G & L guitars - Leo Fender's latest venture.
Great Gibson 335-like neck but with a strat-like body wtih active electronics that can go single-coil or humbucker, active EQ, and I installed an in-out of phase switch on top of it. And a whammy that doesn't go out of tune every minute. I bought it on the spot, and still have it. From what I can tell - G &L is still using pretty much the same electronics, neck, and whammy as my F100 - check it out. It was never quite a Les Paul sound, but could pass for a "Generic Gibson" and "Generic Fender"
But today, you literally have 100s of choices - anything from boutique custom makers to mass-produced offshore stuff that is actually very good for the money. In the '80s I had a little Yamaha I used as a spare, I think it was $275 with case and was a great little guitar with a strat-like sound. I can't remember the model.
But you're right - a Strat is a different animal and takes some getting used to. for one thing it's a lot narrower fingerboard than a Les Paul, and more radius. I never cared for it very much either, but it is definitely strictly a personal preference. If you're thinking of going that way, pick up a Squire - surprisingly good for the money, will give you a feel for it before laying out real dollars.