Help with itinerary

Neuschwanstein and Berchtesgaden are both at considerable distance from München (look at http://www.viamichelin.com/web/Routes) and will cost you the best part of a day.
Opinions on Neuschwanstein differ wildly. It's good to know that it is a nineteenth-century palace built in the style of a idealised medieval castle. The palace at Herrenchiemsee - also nineteenth century - might appeal more to you. It's also much closer to München.
Many find Cologne (Köln) a not very attractive city and the cathedral almost the only worthwhile thing to see.
Haven't you considered wonderful destinations like Heidelberg, Nürnberg and Bamberg as stops on the way from Osnabrück to München? I'm sure you will love them.

You can certainly take a day trip to visit Neuschwanstein from Munich but I think Berchtesgaden is a bit too far. I would do it as an overnight., at least. It's a beautiful drive and there's a lot to do in that area.

Donna

"... we will have 10 - 11 days left to go exploring."

A nice amount of time but not enough IMO to cover so much ground. You just end up in the car (or the train) a long time.

As tonfromleiden says, N'stein indeed is a healthy distance from Munich; it eats up nearly a full day (for a crowded 30-minute tour of a "castle" that isn't a castle at all.) But look at the rest of your trip... check that viamichelin site and you'll see that the ground travel time for just Osnabrück to Cologne to Eltz to Munich will be huge. Then you have Füssen and Berchtesgaden...

I see that Rick Steves' highlights are prominent in your plans; your choices tend to be places with "strong visuals" - places with names that you recognize and that make a big impression in the glossy brochures of tour providers. I would encourage you to see SOME of the those places from the GAS book, but NOT to use the book or his tour route destinations as your planning Bible. In fact you can throw a dart at a map of Germany and find equally interesting and stunning places to visit that will not be so heavily impacted by the international tourist crowd.

Below are some suggestions that take into account what you have mentioned and will save you some transportation time and expense in the general area of Osnabrück, Cologne, and Burg Eltz. Several of them are, like the Cologne Cathedral, UNESCO World Heritage sites. 10 days won't be enough.

AACHEN CATHEDRAL (UNESCO WH)
WW II: Cologne's Nazi Documentation Center (former Gestapo prison); South of Cologne is the Bridge at Remagen museum and across the river is the old-world town of LINZ.
CASTLES: 40 of them, all real ones on a 40-mile stretch of the Rhine River Valley (UNESCO WH) south of Cologne, including Rheinfels and Marksburg
DAY CRUISE on the Rhine from Bingen to Boppard
Rhine villages of BACHARACH and BRAUBACH
Near Burg Eltz: TRIER, Germany's oldest city, UNESCO WH site. Also on the Mosel River: COCHEM and Reichsburg Castle's FALCONRY SHOW
BREMEN is a UNESCO WH site too.
PALACE: see Bückeberg
HAMELN (see tourist info HERE) is just one of numerous towns on the FAIRY TALE ROAD which cuts through Lower Saxony.
The GERMAN HALF-TIMBERED ROUTE connects dozens of impossibly attractive old-world towns.

Jackie - I'll suggest you choose either a northern itinerary (comments above, particularly by Russ) or a southern one. Yes Germany is the size of Montana and is easy to get around with a car but time is spent transiting from place to place and it might be best to focus on either the north or the south.

My wife and I are partial to southern Germany and the Alps, just becuase that's the type of landscape we enjoy. One itinerary would be 3 to 4 days in Munich (possibilities: Viktualienmarkt, Chinesischer Turm Bier Garden, Deutsches Museum, BMW Welt, Olympia Park), then head to Salzburg for 3 to 4 days (Old Town, Fortress, Getreidegasse, St Peter's Cemetery), you can also base out of Salzburg to make a day trip (less than an hour away) to Berchtesgaden/Obersalzberg (Eagles Nest, Berghof, Doucmentation Center).

If you do go to Salzburg I can heartily recommended not driving into or staying in the Old Town but at Hotel Lachenskyhof just out of town and on a conveninent bus route. Wonderful place with an excellent restaurant.

I know you'll have a grand time, no matter where you go.

Prost,

Scott

There was another recent thread that addresses a few of the places that will be somewhat near you when you are visiting your relatives in Osnabrück. Their question was quite different but perhaps you'll see something in the tread that piques your interest.

https://community.ricksteves.com/travel-forum/germany/advice-regarding-german-train-passes