(Car is an '80 but trans/OD & electrics are from an '84)

So what I have found is that when the overdrive is hesitant, if I reach over and give the the underside of the dash a gentle whack with the heel of my hand it kicks in every time.
Obviously I am prodding the relay to go ahead and work but why does it need this encouragement? This sort of solution is only acceptable in the short term. I want it to work properly.
I have examined the guts of the relay under magnification and I can see some fractures in the solder around some of the leads and I intend to try and fix it. But is that alone going to be the cure? It would seem to me like the contact isn't wanting to close and whacking the dash is just enough to get it to go ahead and do so. What's the fix for that condition?

On soldering.
I have looked in the archives and found a lot of useful info but I still think I'm going to f**k something up if I don't set about doing it correctly. Is the object of this excercise to resolder all connections? Or just those with visible fractures? My biggest qualm is my soldering iron. It's your common garden variety 25w pencil with a 5/32 tip. It seems to me that I should be using a tip that more closely resembles an ice pick. What do I need to do here?
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'80 DL 2 dr