First of all you've made a mistake: you need to introduce $N_1$ and $N_2$ so that for any $m_1,n_1 \geq N_1$ you have the property and similar for the other one.
Having fixed that, if you have $|a_m + b_m - a_n - b_n| < 2 \varepsilon$ for $m,n \geq N$, then you are technically done, since $2 \varepsilon$ can be made arbitrarily small by making $\varepsilon$ arbitrarily small.
Typically for aesthetics, you let $\varepsilon$ be the small number for the converging quantity of interest and then choose new small numbers for what contributes to it from there. Here you can take $\varepsilon$ to be your small number for $a_n+b_n$ and then choose $N_1,N_2$ so that you get $\varepsilon/2$-closeness for $a_n$ and $b_n$ respectively. This is valid because the definition that $a_n$ and $b_n$ are Cauchy says you can pick whatever "$\varepsilon$" you want, so you can in particular pick it to be $\varepsilon/2$ (where $\varepsilon$ was already specified).