Can I legally use a trademarked name on t-shirts for sale in a different way than trademarked?

That sort of a thing is always specific to the particular circumstances. You should have an attorney review the details and talk with you about risk if you are concerned. Generally speaking, using a non-famous brand in conjunction with products not associated with that brand in a way that does not cause consumer confusion is not a problem.

Maybe, maybe not. Make your new t-shirt design and then have it reviewed by a trademark lawyer, and seek a clearance opinion before any sales take place. The trademark owner may still object to your use of the mark, and may even be able to convince a judge that your use should be stopped. T-shirts with trademarks on them are often sold by the owner of the mark, or by other authorized sellers.

You are not using it as a trademark or a service mark. Also it would be logical to assume that you have looked at both of these trademarks and you won't be designing anything close in design or color to theirs. Also it would be safe to assume that it would not be scandalous so as to blemish their name. Also it would be safe to assume that your use would not be in any of the same fields as theirs. I would go ahead and design the shirt for them.