I would suggest to get a positive answer to the second question as follows: Let $\{x_i\}$ be a basic sequence in $X$ (existing by Mazur's result) and let $\{z_i\}$ be a rapidly converging to zero sequence in $X$ with dense linear span. Let $\{y_i\}$ be given by $y_i=x_i+z_i$. The sequence $\{y_i\}$ is basic by the result of Krein-Milman-Rutman. The linear span of the union is dense because it contains $\{z_i\}$.

The answer to the first question is negative. It is done by using the construction for the second question, and in addition picking $\{z_i\}$ in such a way that each of it subsequences has a dense linear span in $X$. Such constructions are known, one of them: Consider a sequence $\{u_k\}$ of normalized vectors with dense linear span and let $z_i=\sum_{k=1}^\infty\left(\tau_i\right)^ku_k$, where $\{\tau_i\}$ is a rapdly converging to $0$ sequence of real numbers. Any subsequence of $\{z_i\}$ has a dense linear span because otherwise there is $x^* \in X^*$, $x^*\ne 0$, such that the analytic (in $\lambda$) function $\sum_{k=1}^\infty\lambda^kx^*(u_k)$ would be a nonzero function with infinitely many zeros in $[0,1]$.