We present medium resolution (R = 60,000) absorption measurements of the interstellar Ca II K line observed towards five nearby A-type stars (49 Ceti, 5 Vul, ι Cyg, 2 And, and HD 223884) suspected of possessing circumstellar gas debris disks. The stars were observed on a nightly basis during a six night observing run on the 2.1-meter Otto Struve telescope at the McDonald Observatory, Texas. We have detected nightly changes in the absorption strength of the Ca II K line observed near the stellar radial velocity in three of the stars (49 Ceti, i Cyg and HD 223884). Such changes in absorption suggest the presence of a circumstellar (atomic) gas disk around these stars. In addition to the absorption changes in the main Ca II K line profile, we have also observed weak transient absorption features that randomly appear at redshifted velocities in the spectra of 49 Ceti, 5 Vul, and 2 And. These absorption features are most probably associated with the presence of falling evaporated bodies (exo-comets) that liberate evaporating gas on their approach to the central star. This now brings the total number of systems in which exocomet activity has been observed at Ca II or Na I wavelengths on a nightly basis to seven (β Pic, HR 10, HD 85905, β Car, 49 Ceti, 5 Vul, and 2 And), with 2 And exhibiting weaker and less frequent changes. All of the disk systems presently known to exhibit either type of short-term variability in Ca II K line absorption are rapidly rotating A-type stars (V sin i > 120 km s-1). Most exhibit mid-IR excesses, and many of them are very young (< 20 Myr), thus supporting the argument that many of them are transitional objects between Herbig Ae and “Vega-like” A-type stars with more tenuous circumstellar disks. No mid-IR excess (due to the presence of a dust disk) has yet been detected around either 2 And or HD 223884, both of which have been classified as λ Boötis-type stars. This may indicate that the observed changes in gas absorption for these two stars may not be due to circumstellar activity, but may instead be associated with the stars’ episodic mass loss and passage though low-density interstellar clouds.