Question

1. He decided that she should go shopping.
2. His decision was that she should go shopping.
3. He made a decision that she should go shopping.

4. He suggested that the party be put off.
5. His suggestion was that the party should be put off.
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Can we use all the sentences? Can we omit 'should' as well? Sentences 1 through 3 are the same. Sentences 4 through 6 are the same. Am I right?
6. He made a suggestion that the party should be put off.