Radio-loud active galactic nuclei at $z∼ 2-4$ are typically located in dense environments and their host galaxies are among the most massive systems at those redshifts, providing key insights for galaxy evolution. Finding radio-loud quasars at the highest accessible redshifts ($z∼ 6$) is important to the study of their properties and environments at even earlier cosmic time. They could also serve as background sources for radio surveys intended to study the intergalactic medium beyond the epoch of reionization in HI 21 cm absorption. Currently, only five radio-loud ($R={{f}_{ν ,5\,{\rm GHz}}}/{{f}_{ν ,4400\,\overset{\circ}{\rm A} }}\gt 10$) quasars are known at $z∼ 6$. In this paper we search for $5.5\lesssim z\lesssim 7.2$ quasars by cross–matching the optical Panoramic Survey Telescope & Rapid Response System 1 and radio Faint Images of the Radio Sky at Twenty cm surveys. The radio information allows identification of quasars missed by typical color-based selections. While we find no good $6.4\lesssim z\lesssim 7.2$ quasar candidates at the sensitivities of these surveys, we discover two new radio-loud quasars at $z∼ 6$. Furthermore, we identify two additional $z∼ 6$ radio-loud quasars that were not previously known to be radio-loud, nearly doubling the current $z∼ 6$ sample. We show the importance of having infrared photometry for $z\gt 5.5$ quasars to robustly classify them as radio-quiet or radio-loud. Based on this, we reclassify the quasar J0203+0012 (z = 5.72), previously considered radio-loud, to be radio-quiet. Using the available data in the literature, we constrain the radio-loud fraction of quasars at $z∼ 6$, using the Kaplan–Meier estimator, to be $8.1_{-3.2}^{+5.0}\%$. This result is consistent with there being no evolution of the radio-loud fraction with redshift, in contrast to what has been suggested by some studies at lower redshifts.