$\newcommand\ep{\varepsilon}$ $\newcommand\si{\sigma}$ $\newcommand\Ga{\Gamma}$ $\newcommand\tPi{\tilde\Pi}$ It follows from Theorem 2.1 of this paper or of its better version that for a large class, say $\mathcal F$, of nondecreasing functions $f$, containing the class of all increasing exponential functions, we have $$Ef(X-EX)\le Ef(Y),\tag{1}$$ where $$Y:=\Ga_{(1-\ep)\si^2}+y\tPi_{\ep\si^2/y^2},$$ $\Ga_{a^2}\sim N(0,a^2)$, $\tPi_\theta$ has the centered Poisson distribution with parameter $\theta$, $\Ga_{(1-\ep)\si^2}$ and $\tPi_{\ep\si^2}$ are independent, $$y:=\max_i q_i,\quad \si^2:=\sum_i p_i q_i,\quad\ep:=\sum_i p_i q_i^3/(\si^2 y)\in[0,1],$$ $p_i:=P(X_i=1)$, and $q_i:=1-p_i$. We see that $\ep\in[0,1]$ "interpolates" between the Gaussian and (re-scaled centered) Poisson random variables (r.v.'s) $\Ga_{\si^2}$ and $y\tPi_{\si^2/y^2}$.
From here, one can immediately get exponential bounds on the tails of $X$, or one can get better bounds such as $$P(X-EX\ge x)\le\frac{2e^3}9\,P^{LC}(Y\ge x)$$ for all real $x$, where $P^{LC}(Y\ge\cdot)$ denotes the least log-concave majorant of the tail function $P^{LC}(Y\ge\cdot)$; see Corollary 2.2 and Corollary 2.7 in the linked papers, respectively.
To see better how this works, consider the iid case, with $p_i=p$ and hence $q_i=q=1-p$ for all $i$. Then $y=q=\ep$ and
(i) if $p$ is small then $\ep=q$ is close to $1$ (and $y=q$ is also close to $1$) and hence $Y$ is close to the centered Poisson r.v. $\tPi_{\si^2}$;
(ii) if $q$ is small then $\ep=q$ is small and hence $Y$ is close to the Gaussian r.v. $\Ga_{\si^2}$;
(iii) if neither $p$ nor $q$ is small but $n$ is large then $\ep=q$ is not small and $\si^2$ is large, and hence $\tPi_{\ep\si^2/y^2}$ is close to $\Ga_{\ep\si^2/y^2}$ in distribution, so that $Y$ is close in distribution to the Gaussian r.v. $\Ga_{\si^2}$, just as in Case (ii).
For $f(x)\equiv e^{tx}$ with real $t\ge0$, (1) becomes $$E\exp\{t(X-EX)\} \le\exp\Big\{\frac{t^2}2\si^2(1-\ep)+\frac{e^{ty}-1-ty}{y^2}\,\si^2\ep\Big\}\tag{2};$$ cf. e.g. formula (1.5) in the better version of the linked paper, which implies $$P(X-EX\ge x) \le\inf_{t\ge0}\exp\Big\{-tx+\frac{t^2}2\si^2(1-\ep)+\frac{e^{ty}-1-ty}{y^2}\,\si^2\ep\Big\}$$ for real $x\ge0$. The latter $\inf$ can be explicitly expressed in terms of Lambert’s product-log function -- see the expression in formula (3.2) in the same paper; another useful expression for the same $\inf$ is given by formula (A.3) in this other paper.
