  Dynamical evolution within planetary systems can cause planets to be engulfed
by their host stars. Following engulfment, the stellar photosphere abundance
pattern will reflect accretion of rocky material from planets. Multi-star
systems are excellent environments to search for such abundance trends because
stellar companions form from the same natal gas cloud and are thus expected to
share primordial chemical compositions to within 0.03$-$0.05 dex. Abundance
measurements have occasionally yielded rocky enhancements, but few observations
targeted known planetary systems. To address this gap, we carried out a
Keck-HIRES survey of 36 multi-star systems where at least one star is a known
planet host. We found that only HAT-P-4 exhibits an abundance pattern
suggestive of engulfment, but is more likely primordial based on its large
projected separation (30,000 $\pm$ 140 AU) that exceeds typical turbulence
scales in molecular clouds. To understand the lack of engulfment detections
among our systems, we quantified the strength and duration of refractory
enrichments in stellar photospheres using MESA stellar models. We found that
observable signatures from 10 $M_{\oplus}$ engulfment events last for $\sim$90
Myr in 1 $M_{\odot}$ stars. Signatures are largest and longest lived for
1.1$-$1.2 $M_{\odot}$ stars, but are no longer observable $\sim$2 Gyr
post-engulfment. This indicates that engulfment will rarely be detected in
systems that are several Gyr old.
