Although I can be reasonably confident that you will prefer the revamped New Takanawa Club or Superior rooms to the Non-Club rooms at the ANA IC, I'm not nearly so sure that you will prefer the New Takanawa to the Strings IC.

It's the added Club facilities at the New Takanawa that makes the two such a toss up. The Strings is the better hotel and it's natural to want to spend the end of a holiday at somewhere more 'plush' than one did at the beginning.
But please don't be concerned if you're locked into your decision, I can't imagine your holiday would be ruined as a result.

As for schlepping luggage between the two hotels - there is a shuttle bus between the New Takanawa and Shinagawa station. At Shinagawa station there are two main exits, the Takanawa side which faces West and is the right side for the Prince Hotels and the Konan side which faces East and is the side you need for the Strings.
The Prince Hotels shuttle stops at the taxi bay at the Takanawa side - the service can involve a bit of a wait, so it's easier to use with the help of the bell captain for returning to the station.
I think the Strings will offer an escort and help you with your luggage when you return to the station from there - but I never availed myself of this service. Although I've been here a couple of times, these visits coincided with the time I was in a wheelchair - the walk between Shinagawa station and the Strings is an effortless, smooth, step free glide and our bags stacked beautifully onto the chair, no assistance was required or requested.