Tex,
Well, you are in a bit of a bind aren't you. If you are getting clumps of grass as a result of mowing tall grass, a mulching attachment will not help. When using a mulching mower or attachment, it is necessary to mow MORE often than with a standard deck. If you try mulching taller grass, the deck will fill up with juicy green material and have a tendency to plug. More important, the cut material will not disperse evenly. All manufactureres of mulching attachments that I am aware of recommend mowing when you are cutting only an inch or two of grass and even then the grass must be dry. I often find it impractical to mow that often, so my approach is to use my lawn vacuum when the grass is too tall for standard cutting.
With regard to your current thatch buildup problem, I don't think a core aerator will do the job. You need to first "harvest" the existing thatch. That means starting with some kind of powered dethatcher or rake available from rental shops, or using a small "wheel rake" (estate rake) to bring the thatch to the top so that it can be vacuumed up. I find that it is necessary to vacuum or hand rake up the loose thatch--if you just try to mow it, the mower just blows the lightweight thatch around without pulverizing it as desired.
One manufacturer of the small non-powered wheel rake is Al's Specialties (
www.estateyardcare.com). It is just a miniature version of an agricultural wheel rake used for raking hay. These small units are often available from rental agencies or lawn tractor dealers. I have the small sized unit and like it. It does require adding some weight to the weight boxes. In addition you will probably need to go over the areas with deep thatch several times to get all of it out. However, this rake does the job and is not very expensive. (You can also order these rakes from Northern or other similar catalog operations.)
Once you get the existing thatch out, if you can stay on top of mowing on time (or mowing with a lawn vac) the thatch will not build up very fast. I find that I only need to dethatch about every 2-3 years now.
Good Luck.
JackIL