I have only used the 4-cavity molds, and for those you could use the lee handles, but you'd have to grind or file the handles a little narrower to fit the lyman mold, which would be a bit of work.
You can find used handles and molds on ebay, you can get pretty decent deals for molds+handles...right around $50-$75 is what I see good used 4-cavity molds and handles go for.
Brand new, you are paying about 2x more per mold for the lymans, but I've peened so many Lee sprue plates, and had so many Lee handles fall off my Lee molds, I'm sick of switching parts from mold to mold as parts fail, so in the long run the steel molds are the way to go, for me at least. I'm seriously down to 2 sprue plates that have to be used amongst 5 aluminum lee molds...
You can also rebuild the Lyman molds, so far I haven't been able to find rebuild kits for the Lee molds.
The one thing the Lee molds have going for them is the Tumble Lube design, but other designs will use their Liquid Alox lube just fine, and I had to buy sizing equipment anyways to get those TL bullets to shoot worth a dang.
Edit: I will add that all my problems with the Lee molds have been with the 6 cavity molds, and have mostly been related to the sprue plate wearing to the point that it would no longer cam open, and handles falling off or cracking. Casting wise, they cast fine, I just wish they lasted longer for me. The 2 cavity Lee molds I have so far have been OK.