How to get faster?

At first play the longest time control under blitz and with increment which I think is 5|5. That way you have plenty of time to check your intended move is not a blunder. If this doesn't work simply practice against a computer. Set some winning position on the board and restrict yourself with time, like 3 minutes to win it. Try and try all over again until you make it. Later change the position to another winning one, or a draw or try to play out some opening in short amount of time, it doesn't matter. This is what first came to mind. I hope it will work. After few days you will get used to the speed and evertyhing you did so far will combine in a blitz game.

Also set yourself little goals during the goals so you won't be in big time trouble. For example I will make 15 moves before 1/3 of the time or I won't be more than 45/30/20 seconds behind my opponent. Try to stick and follow those rules, but they don't have to be strict and shpuld be flexible. If you notice your opponent is playing faster than usual ypu should be flexible and let yourself think, it will pay off in that scenario. Also if you feel like there is a tactic or ita a critical moment of the game, you can ignore the rules. As a 1300 you should be able to make solid (doesn't have to be improving move neccesarily) and steady move in less than five seconds in any given moment. If you are still not able to do that you need to practice. Out of 10 tries if you can't make atleast 8 quick moves that are not blunders you should be fine to atleast 1000 blitz. I hope everything I have said helps. I wish you all the best ! :)

Ha, that's the sad truth! The glib answer is that it doesn't really matter why I want to be better at blitz, but honestly I just think it's a fun way to play now and then, and I'm pretty sure it'd be more fun if I blundered less.

As for improving my slow play, that is more important to me than blitz and I spend more time working on it. Currently I'm working through Silman's How to Reassess Your Chess with the goal of improving my positional play, working puzzles to improve my tactics, and playing daily games to put it all into practice and work on reducing blunders.

Finally, I'm not convinced that improving my slow play will have more than marginal effects on my fast play, given the specific nature of my failings as a fast player. No doubt that a rising tide of chess knowledge and experience in my brain will lift both ships, but one of the ships has an unpleasant habit of spontaneously capsizing. And since I'm enjoying both fast and slow play, it makes sense to me to spend a little focused time improving each.

Thanks for the advice and encouragement! I'll try your suggestions.