Bowtie, I've milled a fair amount of short boards out of mulberry, out of what would've been someone's firewood. I cut the main section of the trunk into chunks for bowl blanks---beautiful, BTW. The rest was too crooked to get much more than a couple feet straight, but I sawed up all the 12" and larger dia. stuff into boards, quartersawed it all.
Point I'm making, I didn't seal it, b/c of the time it had already been sawn, but had it in a shed away from direct light; it hardly checked at all. I was very surprised. I would definitely seal your boards, however, as the only reason I didn't seal mine was it had already been cut for some time. FWIW, try to keep both those species out of the light if at all possible. It will turn a deep chocolatey reddish brown, but will retain its beautiful yellow color for a while if its kept out of direct sunlight. I haven't personally worked with osage, only mulberry. I'm under the impression they are somewhat similar, osage, or hedge apple, is harder, and the older darkened stuff I've seen is quite a bit redder than the mulberry I have that's changed to the chocolate-yellow brown.