Thick asparagus stalks are normally considered highly desirable, not at all woody. Have you tried picking them when they are just emerged, only about 8 inches high?

What you have there happens to all crops that come from a crown or bulb. Just like tulips and rhubarb, you need to divide the crown, and that will reinvigorate growth, keeping stalks young and fresh, and will increase your plant populations. Look on google for directions, and wait until the late fall, when the plant has gone dormant before you divide though.

Here is a link that might be useful: Garden

I also have also felt the thicker the stalk the more tender the asparagus. As long as I pick them before they are a foot long and snap off the bottom inch or so I have never had woody asparagus. I may be spoiled though, if we have asparagus in the refrigerator that is 2 days old we wonder what we are going to do with the "old" asparagus. I have never heard of breaking up asparagus roots to rejuvinate them. I am under the impression a bed will last at least 15 to 20 years and then bite the bullet and plant a new bed.

If you break off the bottom where it naturally snaps when bent, then take a potato peeler and peel the outer skin about 1/2 way up it is as tender as it can be.

I have been wanting to grow some Asparagus.
Think I might start this year.

I wish I had your problem, my Mary W. are all small.
This link sales asparagus, but has good information too. Read the how to page, even if you do not want to buy plants.

Here is a link that might be useful: may help