Does it make any statistical sense to subtract / add RMSE from the predicted results to make the model more accurate? If RMSE represents the standard deviation of the residuals (prediction errors), while  residuals are a measure of how far from the regression line data points are, does it make any sense to subtract / add RMSE to the predicted results as some sort of adjustment to make the model more accurate?

Your question indicates that you are thinking about the sampling variability for the parameter estimates in the model that you would obtain from a sample data set.  And, while we cannot make a single prediction "more accurate", your idea of improving the model may be restated as ¿can we add more confidence to our predictions with the rmse?  (If I have completely misrepresented your query, please let me know.)
The answer is:  yes.  And your intuition that it would involve the rmse is essentially spot on.  But, the manner in which we add/subtract this value is a bit convoluted.
I will provide here the formula you would use for a bivariate regression model (one scalar predictor and one scalar response variable).  If you imagine starting with your bivariate regression model, you can make a point-estimate prediction:
$$\hat{y} = \beta_0 + \beta_1 · x_0$$
Now, if you want to add confidence to this prediction, we will add/subtract the rmse...but we also have to account for some of the variability in our estimates for the slope and intercept.  To update the estimate, we have the following formula:
$$\hat{y} = \left(\beta_0 + \beta_1 · x_0\right) \pm t_\text{c.v.}·\text{rmse}·\sqrt{1 + \frac{1}{n} + \frac{(x_0 - \bar{x})^2}{s_x^2(n-1)}}$$
where $t_\text{c.v.}$ is the critical value associated with the degree of confidence you wish to have in your final estimate, $n$ is the sample size, $\bar{x}$ is the mean of the independent variable, and $s_x$ is the standard deviation of the independent variable.
Note, the 1 under the square root is multiplied by the rmse, and this captures the idea that you were suggesting in your post.
I hope this was useful.
