Keeping the dogs out of the garden beds is not a particularly good reason to end up with raised beds! We have two dogs and frequently dog-sit for friends and family. There are sometimes 4 dogs here. We have a large garden and the dogs' needs were considered in laying out the garden. The dogs have never damaged the plantings. One of the primary needs of dogs is to patrol the perimeter of their territory - that usually means they like to run the fencelines! The simple solution is to make a path along the fence and put the garden beds to the inside. As long as there is easy access (e.g. intersecting path or paths) to the fence path, the dogs will usually 'take the path of least resistance' to get to the fence path.

Dogs also like to survey their territory from higher ground - so a raised bed might actually become their preferred spot to park themselves to monitor activity in their territory!

Watch your dogs to see what routes they usually travel around the property and use that information to help plan the garden layout.

Traffic patterns shown on a rough sketch of our backyard garden - note the paths along/near the fences:
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Visiting dog surveying territory from higer ground:
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The best way to keep soil from slipping between the fence slats and keep moisture away ... is to keep soil away from the fence. You can't really place a raised bed directly against the fence without creating a multitude of problems. The proper way to raise grade where fence is involved is to raise the grade first, which might require a retaining wall, depending on your needs. and then place a fence on top of the raised grade.