I'm very much a senior citizen (or OAP) as well, but not quite yet in my dotage. I've always driven while in Scotland (and England), and even at my advanced age, still enjoy the challenge of driving on the left while shifting with my left hand. Those recent models with six forward gears are a blast! However, I can appreciate your reticence regarding driving. My wife refuses to drive in Scotland, so I end up doing all of the driving. We drove over 2,100 miles in July, not including ferry travel.
Having said that, there are lots of places in Scotland which you can visit sans auto. The train service is excellent, although it certainly doesn't go everywhere. Bus service goes to many places in Scotland that you wouldn't normally expect, although a lot of the bus service in more remote areas is based on service to and from schools, with the result that they don't operate on weekends or when schools are closed for hols.
There is an excellent Premier Inn in Inverness, right on the west bank of the River Ness, about a five minute walk to the castle, and a good seven to ten minutes to the train station. There are other Premier Inns in Inverness, but the Nessbank one is closest to the action.
If you're interested in something unique, I'd recommend taking the CalMac Loch Nevis ferry to Eigg or Rum, and staying overnight. You can also tweak the ferry schedule to spend a few hours on Eigg, Rum, or Canna, going out on the Loch Nevis, then returning to Mallaig when the ferry returns to the island on the same day. You can also do a two or three island non landing tour, just to whet your appetite for your next holiday. (And there will be one!) There is really good low level hiking on all three islands. I haven't been to Muck, so I can't vouch for it. Also, there is a small ferry which sails out to the isolated peninsula of Knoydart, which is only accessible by boat or a 16 mile hike in. There's not a whole lot to do on Knoydart, with the exception of hiking on mostly easy roads. But it's a great place to chill out.
You can also take a very scenic and relatively easy (one small rock scramble) hike of about five miles from Bracorina on Loch Morar to Tarbet on Loch Nevis, then take the 3:30 ferry back to Mallaig by way of Inverie on Knoydart. Taxi from Mallaig to Bracorina is about ten pounds plus tip; ferry from Tarbet to Mallaig is 15 pounds per person.
Arran is another good place to visit, with excellent (and relatively inexpensive) local bus service around the island. My wife and I visited there for the first time this year. We've been traveling to Scotland since 1973, and we couldn't believe that we had bypassed Arran until this year. We're already planning to go back.
Best to give John O'Groats a pass. There's not much to see there, and since you're traveling by public transportation, it's not worth the extra effort and expense.
Skye would best be done with a tour, as the local buses mainly serve just the main routes to Kyleakin, Broadford, Armadale, Portree, Uig, and Dunvegan. It would be a real challenge on Skye to visit specific places, then have to wait for the next bus for your onward or return journey. If you need a taxi on Skye, I'd recommend Kenny's Taxis. Skyegirl may have an additional recommendation. We used Kenny's once when we missed the last bus out of Sligachan after a hike in the Cuillins, and had to get back to our car in Elgol.
I can make some recommendations for lodging in most places in Scotland. I'm a budget traveler, but we've always (well, almost always) been able to find good, comfortable lodging at a reasonable price. All of it en-suite. At our age, we appreciate our privacy!
Hope that helps in some way. Best wishes for your plans.