Does my plum tree have canker?

Bacterial canker affects members of the Prunus Genera - plum, cherry, apricot. A swollen area on the bark exudes a light brown gum, usually near the angle of branches. If the swelling encircles the branch it is killed.
Wind or water-borne bacteria enter leaf scars and pruning wounds during the autumn.

* I can't find mention of bark splitting as a symptom, only a cause, ie bad pruning can let infection in

I don't know enough about plums to advise. I've been Googling, but can't find anything relating to bark splitting.

Sometimes bark splits and oozes when we get early and harsh frosts - as we did recently.
If you want to keep the tree, I were you, I'd carefully cut out the affected areas (trying not to spread the presumed infection), then wash the wounds with some alcohol, then paint over with a pruning wound sealant.

If you catch things early, it is often possible to save the tree.
I have a James Grieve apple and for some reason it had several severe cankers appear a couple of years ago.
I thought that it's chances of survival were quite low - it lost about a quarter of it's structure, due to cutting off or cutting out the diseased areas - but it has since recovered and is now doing well.

Planting another plum in the same place could cause problems with establishment - the soil may be full of plum root parasites which could overwhelm a newly planted tree as it tries to establish.
I would plant something else - and plant it several feet away from where the plum is now.

Not all fruit trees will be susceptible to bacterial canker, so you may not need to worry about spraying - only the "stone fruits" get bacterial canker, if I remember correctly. Apples, pears etc are attacked by fungal canker, not bacterial, so the plum poses no risk to those.