I endorse most of Max's recommendations.

Based on my own experience I would definitely recommend being very cautious about RSI, i.e. especially resting carefully, as well as investing in solutions like more ergonomic devices, voice control, reading different resources (e.g. about good posture and different solutions) and visiting physiotherapists and other specicialists. I was largely unable to type or use a computer for 2-3 years due to RSI and I attribute a lot of this not having rested enough early on (despite the fact that I actually did reduce my activity quite dramatically almost immediately upon experiencing symptoms).

Another thing I would note is that although I think it's good to seek help from different experts, I would treat this very critically. I received completely conflicting, but entirely confidently expressed, diagnoses and recommendations from a number of different GPs, physiotherapists and consultant rheumatologists. Some of the literature I read myself also explicitly suggested that tendinopathies tended to be poorly understood by frontline medics, though I'm not in a position to evaluate whether that is the case (or at least true relative to other conditions). Some of the things which were recommended seem to have some evidence suggesting potential for harm (e.g. strengthening exercises, anti-inflammatories and immobilising wrist braces), so there are some grounds for caution.

One of the few things I would recommend that wasn't mentioned in Max's post, so far as I recall, and isn't mentioned in a lot of resources was keeping your hands warm, but I see you mentioned that in your own comment. There also seems to be some evidence that nutrition can be relevant for tendon healing (assuming that your RSI is related to your tendons): see this review. The main things they point to are vitamin C, taurine, vitamin A, glycine, vitamin EA and leucine.

I'm also happy to talk about this 1-1 if you like.

Some things I've tried and found mildly helpful:

• Using an ergonomic keyboard (I use a split keyboard, which also helps with back pain)
• Avoiding typing while feeling cold (this means sometimes wearing a coat at my desk)
• Wearing a wrist brace at night
• Adjusting the height of my desk, and using a desk with sufficient depth so I can rest my forearms on it while typing
• Using my phone with the other hand that isn't affected
• Generally trying to avoid straining movements with the affected hand (during cooking, etc.)

Things I've considered:

• Learning touchtyping with the Dvorak layout (or some other alternative layout) – takes ~20h to learn, benefits seem disputed (academic research and lifehackers claim mixed results). Might look into if things don't get better.
• Using a foot pedal for clicking and modifier keys. Takes some time to learn and set up, though some seemed to like it a lot.

I found using voice dictation on my phone and iPad pretty good, often now I just send emails and messages using my phone instead of my computer.

I find the Google speech recognition on the Google keyboard for Android pretty good, as well as the Apple speech recognition on IOS devices.

One thing I find very useful but I haven't seen recommended anywhere is simply adding a second mouse to your computer. It allows you to easily switch between both hands. This gives your main hand some rest, but doesn't overuse the other one.

Sample size of 1:

After suffering mild RSI in my early 20s, it was completely and permanently resolved by switching to DVORAK for a year, and then back to QWERTY. It wasn't an intentional solution, but forcing myself to re-learn to type from scratch, twice, massively improved my typing habits. 

Googling around on the benefits of DVORAK for RSI offers more anecdotes in either direction, so take this with a whole pile of salt.

Some further recommendations:

• Keep using your hands, acknowledging it may be (partly) psychosomatic, and not worrying too much about it. A friend told me they saw a surgeon for RSI and the surgeon recommended to keep using the hands as normally and not worry too much, and that helped in their case.
• Reducing phone usage; not using the phone in bed while lying down; not playing games on my phone.

I've had RSI in the past, but not from typing, but instead from repetitive motions loading paper into a machine for scanning. I didn't need to see a doctor about it, and addressing it was ultimately pretty straight forward and I was able to keep doing the job that caused it while I recovered. Things I did:

• wore a stabilizing wrist brace to alleviate the strain on my wrist that was causing pain, even when I was not engaged in an activity that would necessarily cause pain
• payed attention to and changed my motions to reduce wrist strain
• rearranged my work so I had more breaks and less long periods of continually performing the motion (I had other job responsibilities so it was easy to interleave breaks from one thing with work on another)

It's now more than 10 years since I developed RSI, and maybe 4 years since I have needed the wrist brace (my need for it rapidly decreased once I left the job). I think never needing it correlated with increased strength, specifically from indoor rock climbing and related conditioning.

I had a very bad time with RSI from 2006-7, followed by a crazy-practically-overnight-miracle-cure-happy-ending. See my recent blog post The “mind-body vicious cycle” model of RSI & back pain for details & discussion.  :)

Small change I found useful: Using my small finger or ring finger instead of the index finger when I have to use my laptop touchpad. This way my wrist is less twisted, which before felt uncomfortable after a time.