Yes, your friend is right. You should have a negative sign in the second line so that ##\Delta V = -\int_R^r E dr##. The overall expression on the right side is positive. Since you are integrating from larger ##r## to smaller ##r##, ##dr## in the integral is negative. So, the integral ##\int_R^r E dr## represents the summation of a bunch of infinitesimal negative quantities. Thus, ##\int_R^r E dr## will yield a negative quantity. Therefore, ##-\int_R^r E dr## is positive.
Going back to the first line, you wrote correctly that $$V_r - V_S = -\int_R^r \vec E \cdot \vec {dr}.$$ Here ##\vec {dr}## denotes an infinitesimal displacement along the radial path connecting the initial and final points. When integrating radially inward from the surface of the sphere, ##\vec{dr}## points radially inward while ##\vec E## points radially outward. Using ##\vec A \cdot \vec B = |\vec A||\vec B| \cos \theta##, the dot product in the integrand is $$\vec E \cdot \vec {dr} = |\vec E||\vec {dr}| \cos \pi = -|\vec E||\vec {dr}|.$$ Here, we used ##\theta = \pi## since ##\vec {dr}## and ##\vec E## have opposite directions. ##|\vec E|## may be written simply as ##E##, where ##E## denotes the magnitude of the electric field field vector ##\vec E##. However, it is wrong to write ##|\vec {dr}|## as ##dr## since the notation ##dr## is reserved for the infinitesimal change in the radial distance ##r## when integrating from ##R## to ##r##. Thus, ##dr## is a negative quantity when integrating inward. The correct relation for this case is ##|\vec {dr}| = -dr##. So, putting all of this tediousness together, $$V_r - V_S = -\int_R^r \vec E \cdot \vec {dr} =+\int_R^r E |\vec {dr}| = -\int_R^r E dr.$$
It is less confusing if you use the rule that switching the limits of integration changes the sign of an integral. (This rule applies to line integrals as well "ordinary" integrals.) So, we have $$V_r - V_S = -\int_R^r \vec E \cdot \vec {dr} = +\int_r^R \vec E \cdot \vec {dr}. $$ Now we're integrating outward so that ##\vec E## and ##\vec {dr}## point in the same direction and ##|\vec {dr}| = dr##. $$V_r - V_S =+\int_r^R E dr. $$