If I want to perform multiple tests, for example, 3 Wald tests (different design matrices), and 2 LTR tests (different full and reduced design matrices) would I need to create 5 separate `dds` objects?

From how I understand your question, the answer is 'yes',

Strictly speaking, you need a different object for each model fit, i.e., for each result of a call to the DESeq function with different inputs, e.g. same data but different model formula, or different arguments for the argument test.

You can call the function results on the same object with different arguments (e.g., contrast) to perform different tests.