Exterminator

If you can see direct evidence of an infestation, you may have a fairly serious problem, and you should call an exterminator. Caulking damaged sites probably does fairly little, as termites work internally for the most part, only making "mud tunnels" (a composite of digested wood and a liquid compound the bugs secrete) to move from one place to another above ground. It is also possible that what you are experiencing with the dead bugs is what is commonly referred to as a "bloom", wherein a colony - often in April or early spring - launches thousands of its "breeders" above ground, which, in turn, try to find new places to establish colonies. When the place they land is your basement floor, they often die off quickly, not finding new soil to borrow down into. But a bloom is a sure sign of an infestation, regardless of severity. I am not an exterminator, but did a lot of research on this when I had a mild infestation several years ago. You might contact Kingsway Exterminators, a firm that successfully resolved our problem, but bear in mind that you need to hire someone that offers a guarantee of their services - and at least for 5 years in terms of termites. This is especially important since most of the damage these bugs can do will never be visible outside your walls. Good luck.

Have the seller pay for a reputable termite extermination company to do their thing and certify there is no termite issue.

What did they pay for this 4-fold inspection?

Geting peppermint flakes you won't ever see a mice ever again.