Yep, I definitely feel the same way - but I think there's hope. I've found that it's best for me to save a list of the articles and books I want to read, just like you suggested. (Personally, I like to keep separate lists for work-related reading and leisure reading.) Then, at the end of the week, I'll set aside a day or even just a few hours for reading time.
During my reading time, I first just read through the abstracts of those papers - a well-written abstract gives enough information to give me an idea of what's out there without getting me into the nitty-gritty details, and I'm able to get through lots of abstracts in a short time (1-5 minutes / abstract). Then, for the abstracts that caught my eye, or where I want a better explanation of what the paper's about, I'll go back through the introduction and discussion (this might take 5-10 minutes / paper). Finally, for a few papers that seemed especially relevant, I'll go back and read through the methods and results to really understand what they're doing. This can take a while (30 minutes - several hours / paper), so often I'll only look at one or two papers in this final step.
This approach gives me a high-level overview of what's out there, and at the same time allows me to get into the details of a few papers that most interest me. I hope that over time, I'll get to the point where I've thoroughly read and understood many of the well-known papers in the field.
For academic nonfiction, I'll take a few hours on Friday afternoon to read something work-relevant. I try not to approach this from the standpoint of "how many books can I finish?" Instead, I try to just read a chapter or a section that I find interesting, and sometimes this naturally leads me to go through whole books over time.
News & Leisure
That leaves me a bit of time in evenings or on the weekend to do a bit of light fiction reading if I'm in the mood for it. If I'm not in the mood for it, then I just let it go; no point in reading to relax if I don't even feel like reading in the first place.
For news, I try to take about 10 minutes or so each day, either when I'm eating breakfast or when I'm standing in the lunch line, to read a news site (I use Reuters) to see what's happening. I find this keeps me reasonably well-informed about current affairs.
The biggest challenge for me is to avoid getting bogged down with the feeling that I'm falling behind in reading. Although there are thousands of papers and books that I would enjoy, there's just no way that I can churn through tens of papers per week and still truly understand each individual paper. It's a process, and I've accepted that it will take time. I try not to think of my reading list as a "backlog"; I try to think of it as a list of ideas for my next reading time.
Obviously, there are many other ways to manage reading, so in the end it just depends what works best for you. I'd suggest experimenting with a few different strategies until you find one that works well. Let us know when you come up with something good!