General considerations
I think it will help to carefully write out all the expressions. The PDE should be read component-wise
$$
\partial_{tt}\psi_j(x,t)-\nabla^2\psi_j(x,t)=F_j(x,t)
$$
Where $x\in\mathbb{R}^3$. I've replaced $J$ with a generic source $\mathbf{F}$, $\mathbf{E}$ with a generic field $\boldsymbol{\psi}$, and set all constants to unity. The causal Green's function is
$$
G(r,t)=\frac{\delta(t-r)}{4\pi r}
$$
The solution for $\psi_j$ is
$$
\psi_j(x,t)=\int dt' \int d^3x' \ G(x-x',t-t')F_j(x',t') \ \ + \ \ \text{surface terms}
$$
The surface terms are to match initial conditions which we can specify to vanish (see eg. Zangwill Modern Electrodynamics chapter 20). The delta collapses the $t'$ integral and we are left with
$$
\psi_j(x,t)=\frac{1}{4 \pi}\int d^3x' \ \frac{F_j(x',t-|x-x'|)}{|x-x'|}
$$
Note the arguments of the source appearing in the integrand.
The wave equation for $E$
Because $J$ and $\rho$ form part of a continuity equation, they cannot be specified completely independently. For $\psi=E$, the source should be $F_j=-(\partial_t J_j+\partial_j \rho)$, not just $\partial_t J$. You can check this by deriving it from Maxwell's equations. The expression for $E$ is
$$
E_j(x,t)=-\frac{1}{4\pi} \int d^3x' \ |x-x'|^{-1} \bigg[\partial'_{j} \rho(x',t-|x-x'|) \ + \ \partial'_{t}J_j(x',t-|x-x'|) \bigg]
$$
Note the primes on partial derivatives within the integral: the retarded time is $t_r:=t-|x-x'|$. You can see why it's preferable to write the integral as
$$
E_j(x,t)=-\frac{1}{4\pi} \int d^3x' \ |x-x'|^{-1} \bigg[\partial'_{j} \rho(x',t') \ + \ \partial'_{t}J_j(x',t') \bigg]_{t'=t_r}
$$
With $R_j:=x_j-x_j'$, and using the multivariate chain rule we find (Jackson chapter 6.5)
$$
\big[\partial'_j \rho(x',t') \big]_{t'=t_r} = \partial'_j \big[ \rho(x',t')\big]_{t'=t_r} - \hat{R} \big[ \partial'_t \rho(x',t')\big]_{t'=t_r}
$$
On integrating the term $|x-x'|^{-1}\partial'_j \big[ \rho(x',t')\big]_{t'=t_r}$ by parts, we find Jefimenko's equation for $E_j$.
Finally, if you want to specify a thin wire of arbitrary shape, the expression is not as simple as you may think. Start by parametrizing in $\tau$ such that the wire is the set of points $(x,y,z)=(p_1(\tau),p_2(\tau),p_3(\tau))$, then (up to a proportionality constant)
$$
J_j(\mathbf{x})= \int d\tau \ \frac{dp_j}{d\tau} \delta^{(3)}(\mathbf{x}-\mathbf{p}(\tau))
$$
If there is a single monopole it does not matter whether $A$ is real or not but if you have two or more sources then their relative phases, and thus the phase of $A$, do matter. The same holds if the source is distributed and not point-like.
Acoustic phase is important if you care about spatial distribution of the wave field, ie., radiation pattern, but to a first (?) order human hearing is not sensitive to phase. In other words humans cannot hear the difference between $x_1(t)=K_0cos(\omega_0 t+\phi_1)$ and $x_2(t)=K_0cos(\omega_0 t+\phi_2$. Apparently, humans can "hear" $\omega_0 $ (frequency) and $|K_0|$ (amplitude) but not $\phi$ (phase). In other words, the human ear is a frequency sensitive non-coherent intensity detector, just like a spectrum analyzer. This is true whether the oscillation be damped or undamped.
A phase sensitive detector is really two frequency coherent detectors in "quadrature", that is phase is measured relative to two reference signals one is, say, $r_c(t)=Rcos(\omega_0 t+\psi)$ the other is $r_s(t)=Rcos(\omega_0 t+\psi - \pi/2) = Rsin(\omega_0 t+\psi)$ where $\psi$ and $R$ are the arbitrary phase and amplitude of the local reference, resp. While measuring the time evolution of $x(t)=K_0 cos(\omega_0 t+\phi)$ relative to the oscillations of $r_c(t)$ and $r_s(t)$ we can calculate $\phi - \psi$ and then all the other relative phases between the other sources, as well, at the same frequency. Human and I think most if not all animal hearing is very sensitive to time delay between two sources, this is how we discern the direction from which the sound is emitted. It is amazing how sensitive the directional hearing of grasshoppers or crickets is despite the fact that spatial separation between the "ears" of a grasshopper is at most a few millimeters. One wonders if the cricket's hearing may be phase sensitive then, but I am just guessing here and do not really know.
Best Answer
The numerical integration of hyperbolic partial differential equations (PDE) is apparently a direct extension of the usual discretization algorithms used in the case of ordinary differential equations (ODE) and of the related stability analysis. However, the analogy is not complete. The mathematical theory behind PDE introduces some additional constraints, at least in the case of explicit methods, that are not present in the case of ODE.
This is the case of the so-called Courant–Friedrichs–Lewy (CFL) condition. The physical motivation and general description of the condition go as follows.
Hyperbolic PDE's are characterized by a set of conic-like hypersurface passing through each space point (a couple of lines, in the simple case of one space and one time variables) determining which points in the past can causally influence the given point and which points in the future can be influenced by the same point. It is the familiar light-cone concept of Special Relativity, whose origin is actually in the wave equation controlling the light behavior. Courant, Friedrichs, and Lewy were able to show that a necessary condition for the convergence of an explicit numerical method for hyperbolic PDE is that the physical light-cone must be contained in the numerical light cone.
In the case of your scheme, the condition requires that. $$ \frac{c \Delta t}{\Delta x} \leq 1, $$ which is not the case, with the data you have reported.
Therefore, if you play with the simulation parameters ($\Delta t$ and $\Delta x$) to satisfy the CFL condition, it should be possible to get the correct numerical behavior of the one-dimensional wave.