I don't think age-dating oceanic crust is too common, but I do believe the
Ocean Drilling Program has dated some cores - try searching on their website (sorry, don't have time to do this myself). Also, there are a lot of papers on the correlation of biostratigraphy, stratigraphy, and age-dated terrestrial basalts with oceanic basalts that also constrain the ages of polarity chrons.
Turning to oceanic crust drilling, the Ocean Drilling Program has certainly drilled deeper than 105'. In fact, they have gone as far as 2.1 km (6,500+ feet) and possibly further. Last I heard, the deepest oil exploration hole in the ocean was about 3 km (9,800+ feet). I don't know if that still stands, however. See the following links for some deep holes.
talk of going 4,600 deep (Sept 2003 press release)
Hole 504B: penetrates 2,111 meters of oceanic crust (year???)
I understand why you think the magnetic reversals wouldn't work, but I can assure you, scientists have considered and solved the question you have, as well as many others. Check out these sites for some info on magnetic reversals:
reversals and tectonics - check out the other links for more information.
also
from the USGS
There is a lot of info on the web about this. Take a look around.