(I suspect I didn't wipe off enough of the oil or perhaps it didn't have time to dry before the pouring rain)

Yes and yes. The finish you used is a penetrating oil. The proper application steps are:

1. Apply a heavy coat to flood the surface
2. allow to dwell for several minutes
3. wipe of ALL excess before it starts to dry
4. Buff with a clean rag after wiping
5. Allow at least 24 hours to dry before getting wet

Get a gallon of mineral spirits (low odor works fine) and promptly remove all the failed finish. Get rid of the gum, then get another clean rag soaked in mineral spirits and rub out the entire finish to remove all surface residue. Act fast before the gum hardens. Put furniture out of the rain until you fix this problem. Wait a few days before attempting to re apply the product. Don't put any more product on until you get all the failed finish off the surface. Some of the finish you applied will have penetrated into the wood, this is ok, just make sure you get everything off the surface.

Old bed sheets are the best rags for this. T shirts are the next best choice. Don't use anything that will transfer lint or you'll have another mess to fix.

Aidan M - Thank you!! That is great advice and I will definately give it a try.