First, I don't have this inability to proces verbal conversations, but I always end any meeting or conversation that contains new tasks or software requirements with a request for the person who added the task to follow-up in writing in the appropriate system. It is not a real requirement until it is in the tracking system or the project management system or in an email (depending on what the new task is). I bring this up because people are not automatically going to think there must be something wrong with you if you ask for things to be confirmed in writing. So you can solve a good part of this problem by doing that without mentioning that you have a hard time processing.
Next, people are going to continue to communicate verbally and that will be true of most places where you could work. So you need to think what you can do to minimize the issue.
One thing is to look for workplaces where you can work remotely or where a large percentage of workers are remote. People seems to adapt better to doing more things through formal systems when everyone on the team is not readily available for a quick meeting.
Another thing is to volunteer to take notes for the meeting. Because you are the formal notetaker, if they go too fast, you can just say, "Hey wait I'm still on point 1. Now was else did you say?" Because they know that formal note taking slows people down, no one will find it odd if you ask them to repeat to make sure you get it right. In a meeting room, you could be the person who writes down the suggestions as they are made on the whiteboard. The others will see if you missed anything important and ask you to add it to the whiteboard.
Another thing to do is to record (but there are legal issues, so make sure it is ok and that people know in advance that they are being recorded). In this case, you will need to let them know that it is because you need to listen to the conversation several times not to document wrong doing. So using this technique will mean you will have to disclose. I would use it as a last resort.
You could also simply always carry a paper notebook and take written notes. It has been shown that handwriting accesses a different part of the brain than typing and retention of information is far greater when you handwrite notes. Since retention is your problem, have you tried handwriting notes rather than using a laptop or tablet? The activity of taking notes also engages your brain in the conversation more than just listening. Depending on the type of disability you have, this might not work for you, but if you have only ever just tried to listen to and then later remember conversations or only taken notes by typing them, it could make a difference.
I leave you with the idea that disabilities are just things to learn to overcome. Everybody has something they don't naturally do well. Many more people than you are aware of have diagnosed disabilities. It is often their attitude towards overcoming their disability that ultimately dictates whether it is career-limiting or not. I have worked with a deaf auditor who had to interview strangers for a living, a dyslexic person whose job was to perform Quality Control on large technical documents and he was so successful that he was promoted to the head of the QA department. (In fact I knew him for 8 years before I found out he had a learning disability.) Pretty amazing for someone who had difficulty reading and writing. I have worked with depressed people who still manage to go to work and work well even when they are depressed and people in wheelchairs for a variety of reasons with successful careers. So don't give up on finding ways to adapt.
