The simple answer to your question is yes, you should close the slide out on your park model trailer before you move it.
If you attempt to move it even a short distance with the slide out open, you risk causing major damage to the slide out. The slide out structure is not designed to withstand the extra stresses that it would be subjected to if you leave it open while it is being moved.
The time you spend moving furniture out of the path of the slide out and closing it could save you hundreds if not thousands of dollars in repair bills that could occur if you attempt to cut corners by leaving it open.
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