I forgot to mention that both trees have been producing apples for 3 years--just a few on each tree in 2017, more in 2018, and both had good crops in 2019.

Clarice

GH said
My guess is that it's anthracnose.  Other, more experienced, members may have another suggestion or better idea.

My Akane on M.26 rootstock developed anthracnose a couple of years ago, and the damage on your trees looks similar.   I cut out at least half of the tree, then used my version of tree paste on the wounds and infected areas that I couldn't remove.  Last summer it produced a couple of dozen apples, and there were no additional diseased areas; but this summer should give me a better idea if it's going to survive.

My biggest concern was (should have been) in preventing the spread of the disease to other trees.  So far that hasn't happened, and I add tree paste periodically to ensure that the wounds stay covered.  In retrospect the Akane probably should have been removed immediately and the diseased wood destroyed.  It seems to me that this may be more of a gamble than it's worth - one tree versus how many more trees in my orchard?

If you want to try the tree paste instead of immediately cutting down your trees, this is my modified version:  compost, coarse sand, and Surround or another clay, mixed with just enough water to form a paste.  On a rainless day coat every diseased area or wound with the paste.

I hope that more experienced orchardists will weigh in with their opinions, both in what they think it is and also how best to deal with it.  

Thank you for your response.  If the trees are afflicted with anthracnose, it's a pretty difficult thing to get rid of, isn't it?  When do you recommend I try cutting out the damage?  I'm assuming after our spring rains are over?  I would like to at least attempt to salvage the trees, since we're really happy with the apples the trees have produced so far.

I read about cauterizing trees online.  It might be worth trying on one of the trees at least.

Thanks for chiming in!