It seems to be that when a new domain is being used, and in time, you will want to delete the older domain. It always seems to be the case at some point for everyone. For that reason, I will make the assumption that you will eventually want to make the same decision even if you do not. It does not hurt thinking and working toward this goal.
I appears that you have two things going for you: one, in effect, a domain name transfer even if it is not exactly the scenario; and two, at least one property with significant value. This is a perfect scenario for your situation. It is workable.
The first thing I would recommend is doing an audit to see what is valuable in the old domain. By that, I mean pages that return traffic that you want. You can begin there. If it is possible, you can update these pages for the new site. Just try and make sure they are not too much of a duplicate. It is often good just to do a rewrite of the same topic paying attention to what it is that users found valuable and how they found the page through search and links. Often, you will imagine new ways of adding value and covering the topic better. I would be using a spreadsheet for this. I know, it sounds like work- bummer! But it really does not have to be too difficult. You do not have to post the pages right away. In fact, I would wait and make all of the significant changes at once or as close as possible.
The second thing I would do is evaluate the link profiles of both sites. I would identify the links that you likely can change, any links you will want to replicate, and any links you can add to the new site. The reality is this- even with a 301 redirect, any new site has to stand on it's own eventually. It may be wise to build the link profile of the new site to match or excel the link profile of the older site. I would build as many links as possible and work the links you could change once the appropriate updated page exists.
Once I felt I had a good view of the situation, I would add value to the new domain that does not exist on the old domain. Go ahead and get this out of the way.
The next thing I would do is set-up 301 redirects from the most valuable pages of the old domain to the rewritten pages on the new domain (posting them as I go) and if there are links I can get changed, I would have as many of them point to the updated page. I would also seek new links to the updated page on the new domain where I can as much as I can. You can do this one page at a time or to groups of pages, but I would pay attention to the most important pages first. This would preserve any link value from the old page to the new page. I would do this for as much content as I could over a period of time. The idea is to begin the transfer of value onto the new domain. For the other less significant pages, you can possibly create a link to the new site where it makes sense for the user. Do not get too carried away and make sure that the topics are related. Slow and steady wins the race in search.
Once this is done and as much value has been passed as possible, and the link profile of the new domain is strong paying particular attention to the updated pages, then you can make the decision to potentially delete the old domain. In this case, you will do things a bit differently for a while. I would then be doing a blanket redirect of the old domain to the new domain and then redirecting on the new domain, the URIs of the important pages to the updated pages and making sure that I handled redirects of the lesser important pages to somewhere that makes sense. Again with the user in mind. It may be that you will need to create some temporary landing pages, but again, do not get too carried away. Once this has soaked in for a bit, you can decide how to boost the performance of the new site further and do that. It will be clear at this stage.
After this has existed for a while and you feel confident that you have done as much as you can to recreate the value on the new domain, you can remove the old domain if you want.
There will always be a shift in the SERPs at each stage and it will take time to soak in. I would take this one step at a time and simply wait till things right themselves before taking any more steps. The idea is to as painlessly as possible, transfer the value of the old domain to the new even if you do not want to delete the old domain. You will be better off doing this regardless. It will give you the opportunity to evaluate what has been working and replicate that success elsewhere. Regardless, you want your new domain to perform as well as possible on it's own. Once you are satisfied that you have squeezed the last bit of juice from your efforts, then you will have confidence to delete your old domain if you want to. In fact, it may pay to delete it eventually.