Oh no! This isn't looking good! 

I'm seeing signs of a disease called bacterial canker. You can read more about it in OSU's PNW Pest Management Handbook. The globs of sap are one sign. Look closely. You'll also see dieback of the twigs and flowers plus some signs of cracking in the bark. 

How long ago did you plant this tree? If it was this spring you may want to contact the nursery to see about a possible replacement. 

Wow!  Not what I expected.  Is the canker mostly transmitted through the ground and roots system, or through  the air?  The handbook doesn't paint a glowing picture of recovery, so is it better to spray and wait or remove?  Will the canker spread to other non-plum trees?  The trees are in their third year, grafted on a plum rootstock..  Hate to get rid of the tree, but may have to look for a different graft.

The bacteria that causes this issue typically spreads in splashing rain, irrigation water, or even on tools (could be spread by pruning shears, for example).

What types of trees do you have nearby? Are they also Prunus sp (cherry, peach, etc.)

Yes - I have a peach nearby and different plum in the vicinity.

I'll add that my older trees do not show this symptom - just the new ones I planted.

If you decide to leave them in, then be very careful when moving between the trees. You don't want to inadvertently spread the bacteria to the unaffected trees. Any time you are creating a wound is especially vulnerable. So pruning, thinning and picking. 

Work the unaffected trees first and then move to the diseased ones. Consider having two sets of pruners or giving yourself time to sanitize equipment between trees. Here's a short article comparing the different sanitizing solutions used. 

Hi again,
Were the peaches sprayed with anything this winter or spring?

At least some of this looks like the start of shothole disease. This is a fungal disease that affects the leaves. The affected parts of the leaves fall out, leaving a shothole appearance-hence the name. Here's info on managing peach problems. 

Looks like you might be losing leaves prematurely? Over time that will affect the vigor of the tree. Consider making treatments this fall and winter to help the tree next growing season.