Considering two function series i know that $$\sum_{n\geq 0} f_n(x) \mathrm{\,\, converges \,\,\, absolutely \,\,\, and } \sum_{n\geq 0} g_n(x) \mathrm{\,\, converges \,\,\, absolutely} \implies \sum_{n\geq 0} (f_n(x)+g_n(x)) \mathrm{\,\, converges \,\,\, absolutely}$$
But can anything else be said about the absolute convergence of $\sum_{n\geq 0} (f_n(x)+g_n(x))$ if $\sum_{n\geq 0} g_n(x)$ or $\sum_{n\geq 0} f_n(x)$ or both converge only conditionally (that is, converge but not absolutely)?
Can $\sum_{n\geq 0} (f_n(x)+g_n(x))$ converge absolutely in that case?
It is possible to analyze possible behaviors using constants (instead of functions).
So given $f_n$ and $g_n$ such that both $\sum f_n$ and $\sum g_n$ converge conditionally, you are asking if it is possible that $\sum (f_n + g_n)$ converges absolutely.
The answer is yes. For instance, consider $$ \sum_{n \geq 1} \frac{(-1)^n}{n} + \sum_{n \geq 1} \frac{(-1)^{n+1}}{n}.$$ Each series converges conditionally, but their termwise sum converges (trivially) absolutely to $0$.
More generally, if $f_n = -g_n$, then this will occur. If $f_n = -g_n + h_n$, where $\sum g_n$ is conditionally convergent and $\sum h_n$ is absolutely convergent, then $\sum f_n$ is conditionally convergent and you will also have $\sum (f_n + g_n)$ is absolutely convergent.