How can the particle velocity field due to an acoustic monopole be derived in terms of pressure?

I have seen in numerous literature sources (e.g. this chapter from MIT open courseware, pages 414-415) the following method of relating the pressure to particle velocity. Begin with the pressure field of an acoustic monopole (point source with uniform spherical waves):
$$p(r) = frac{A}{r}e^{j(omega t – kr)} $$
where $r$ is radial distance from the source, $A$ is some arbitrary amplitude, $omega$ is the angular temporal frequency of the signal, and $k$ is the angular spatial frequency.
Newton’s second law (as described in the literature) relates particle velocity $textbf{u}$ to the pressure gradient:
$$rho frac{partial textbf{u}}{partial t} = -nabla p$$
Radial symmetry in this case means that the pressure gradient is dependent only on $r$:
$$nabla p = frac{partial p}{partial r} textbf{e}_r = left[ -frac{A}{r^2}e^{j(omega t – kr)} – frac{A}{r}jke^{j(omega t – kr)}right] textbf{e}_r$$
$$nabla p = (-frac{1}{r} – jk)p textbf{e}_r$$
Newton’s second law can therefore be written:
$$rho frac{partial u_r}{partial t} = (frac{1}{r} + jk)p $$
Assuming an oscillatory solution for $u_r$, $dot u_r = jomega u_r$, which gives
$$ rho j omega u_r = (frac{1}{r} + jk)p $$
Dividing through by $jk$,
$$ frac{rho omega u_r}{k} = (frac{1}{jkr} + 1)p $$
$$ u_r = frac{1}{rho c} (1- j frac{c}{2pi fr})p$$

That’s the result that I see wherever I look, but had originally taken an approach in which I considered a differential spherical element of volume $textrm{d}V = r^2 sin(theta)textrm{d}r textrm{d}theta textrm{d}phi$.

A force balance in the $hat r$ direction on the element yields
$$ Sigma F_r =
left( p – frac{partial p}{partial r} frac{textrm{d}r}{2} right)
left[ left(r- frac{textrm{d}r}{2} right)^2 sin(theta)textrm{d}theta textrm{d}phi right] –
left( p + frac{partial p}{partial r} frac{textrm{d}r}{2} right)
left[ left(r + frac{textrm{d}r}{2} right)^2 sin(theta)textrm{d}theta textrm{d}phi right]$$
Dividing through by $sin(theta)textrm{d}thetatextrm{d}phi$ and distributing terms,
$$ frac{Sigma F_r}{sin(theta)textrm{d}thetatextrm{d}phi} =
left( p – frac{partial p}{partial r} frac{textrm{d}r}{2} right)
left( r^2 – r textrm{d}r + frac{textrm{d}r^2}{4} right)-
left( p + frac{partial p}{partial r} frac{textrm{d}r}{2} right)
left( r^2 + r textrm{d}r + frac{textrm{d}r^2}{4} right)
$$
$$ =
pr^2 – prtextrm{d}r + p frac{textrm{d}r^2}{4} –
frac{partial p}{partial r}frac{textrm{d}r}{2} r^2 +
frac{partial p}{partial r}frac{textrm{d}r}{2} rtextrm{d}r –
frac{partial p}{partial r}frac{textrm{d}r}{2} frac{textrm{d}r^2}{4} $$
$$
– pr^2 – prtextrm{d}r – p frac{textrm{d}r^2}{4} –
frac{partial p}{partial r}frac{textrm{d}r}{2} r^2 –
frac{partial p}{partial r}frac{textrm{d}r}{2} rtextrm{d}r –
frac{partial p}{partial r}frac{textrm{d}r}{2} frac{textrm{d}r^2}{4} $$
Neglecting higher order terms, this force balance reduces to
$$
frac{Sigma F_r}{sin(theta)textrm{d}thetatextrm{d}phi} =
-2prtextrm{d}r – frac{partial p}{partial r} r^2 textrm{d}r
$$
Dividing through by $r^2textrm{d}r$, the sum of the forces (in the $hat r$ direction) has been divided by the differential volume
$$
frac{Sigma F_r}{r^2sin(theta)textrm{d}rtextrm{d}thetatextrm{d}phi} =
-frac{2p}{r} – frac{partial p}{partial r}
$$
The discrepancy between the result of this approach and that of the literature is clear at this point. Moving on, recalling that
$$ frac{partial p}{partial r} = (-frac{1}{r} – jk)p $$
gives the equation of motion for the fluid due to the acoustic monopole
$$ rho frac{partial u_r}{partial t} = -frac{2p}{r} + frac{p}{r} +jkp$$
$$ rho j omega u_r = left(-frac{1}{r} + jkright)p $$
Rearranging, we arrive at my original result:
$$ u_r = frac{1}{rho c} (1+ j frac{c}{2pi fr})p$$
And compare to the solution that I find elsewhere:
$$ u_r = frac{1}{rho c} (1- j frac{c}{2pi fr})p$$

It’s easy to look at this inconsistency and think of it as a sign error, but I don’t think that’s the case. I suspect that the problem begins with the formulation of the equation of motion. All external sources that I have seen have taken as given
$$rho frac{partial u}{partial t} = -nabla p$$
but is this really correct for a spherical wave, in which the “front” and “back” surfaces have different areas on which the pressure acts? This expression looks like a plane wave solution to me.

Where in my derivation have I gone astray? Many thanks to anyone who attempts to clarify this concept for me!