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Results from 11 years of radio timing for eclipsing black widow millisecond pulsar (MSP) binary, J1544+4937, is presented in this paper. We report a phase-connected timing model for this MSP, using observations with the Giant Metrewave Radio Telescope (GMRT) at multiple frequencies and with Green Bank Telescope (GBT). This is the longest-duration timing study of any galactic field MSP with the GMRT. While extending the timing baseline from the existing 1.5 years to about a decade we report the first detection for a significant value of proper motion ($\mathrm{μ_{T}} \sim$ 10.14(5) $\mathrm{mas/year}$) for this pulsar. Temporal variations of dispersion measure ($\mathrm{ΔDM~ \sim 10^{-3}}$ pc $\mathrm{cm^{-3}}$) manifested by significant determination of 1st, 2nd, and 3rd order DM derivatives are observed along the line of sight to the pulsar. We also noticed frequency-dependent DM variations of the order of $\mathrm{10^{-3}~ pc~ cm^{-3}}$, which could arise due to spatial electron density variations in the interstellar medium. This study has revealed a secular variation of the orbital period for this MSP for the first time. We investigated possible causes and propose that variation in the gravitational quadrupole moment of the companion could be responsible for the observed temporal changes in the orbital period.