It hard to say without seeing the structure but it sounds like the decking was under engineered for the weight of asphalt shingles to begin with. You could put the OSB overtop but by the time, you clean that surface up (pull nails, etc) and adhere the new surfaces together, you might as well strip it.
You're also leaving extra weight on the roof and if you are already feeling degradation between the rafters, you may ultimately end up with the same problem just thicker.
If you are going to go with asphalt, you might want to upgrade to at least 7/16th OSB depending on your rafter spacing.
The plywood of your roof is typically not as thick as your standard galvanized steel roofing nail, so it is expected for nails to show through this in your attic. The plywood can and is expected to take a bit of damage over time and this is okay as long as the shingle has something firm to hold itself down onto.
Over time if the plywood is too damaged to properly hold down new shingles or becomes too dangerous to walk on safely then a roofer should be able to identify such compromised plywood board and replace it as needed.
What is worrying about your question is that you specified that the roofing nails have split and damaged the roof trusses. Standard roofing nails should not be long or thick enough to split or crack structurally sound dimensional lumber.
If this is what happened then you may have a problem unrelated to the quality of the roofing job. Your home is over 50 years old in a warm wet climate, so you should inspect for a number of different problems.
Dryrot
Termites
Carpenter Ants
If you have any of the following indications of current or past damage to any of the three items then that would explain split, cracked or damaged roof trusses during a typical roofing job.
On another note: The soffit typically fits into a groove that is known as J Channel when installed on a roof overhang. The roof overhang is typically MUCH too tall for any roofing nails to reach soffit. I imagine instead you might be seeing a roofing nails perforating the fascia, which can be a sign of a sloppy rushed job. Without pictures though we can't tell that for sure.
Best Answer
So your existing roofing is either wood shingles, wood shakes or tile ( clay, concrete, etc.) These types of roofs can span across gaps in the roof sheathing.
Your new roof is going to be composition shingles that need solid sheathing.
Yes, it is quite common to add solid sheathing over the top of the spaced sheathing. The added weight will be off-set by removing the heavier existing tile roofing and the heavier wood shingles and shakes (when they are soaked with water).
Here are some precautions: 1) make sure the new roof sheathing is a minimum of 1/2” thick. Roofing manufacturers require a minimum of 1/2” for “holding power and withdrawal”...blow off. 2) Increase the height of your perimeter flashing by at least 1/2” to compensate for the additional height due to the thickness of the roof sheathing. (You want the same lap over the siding.)