Mean Highest High Water is a reference surface corresponding to the height of high tides, averaged over a certain period of time (usually 19 years, a value close to a Metonic Cycle and a Lunar Node cycle). Data from tidal gauges, satellites, etc. are usually processed and stored in grid files, which contain the difference in elevation between different height datums for each cell of the grid, and can then be used to convert from a height datum one to another.
Some tools like VDatum come with these files built-in, and can be used to transform between several Vertical Datums including MHHW, NAVD88. The software contains grid files for several height reference surfaces, like geoids, the height of tides, etc. and can convert from tidal to orthometric or ellipsoidal datums. For example, you can convert your Projected Sea level rise raster from MHHW to NAVD88, and the resulting raster will have the projected mean high water elevations expressed in NAVD88, which can then be compared to another DEM in NAVD88 to create a map of the extent of mean high tides in the future.
Behind the scenes, what happens in that VDatum will use the grid file containing the height of MHHW relative to Local Mean Sea Level (LMSL), then will use the grid containing the height of NAVD88 relative to LMSL, in order to perform the correct transformation.
Because the Earth is dynamic, everything changes over the years. Sea level changes, the crust moves up or down, so any reference surface related to the ground or the sea move relative to each other over time, up to several centimeters a year, so when working with high accuracy data, it becomes important to know the Epoch in order to perform accurate transformations. Depending on the desired accuracy of your analysis as well as the accuracy of the grid files in VDatum, this may or may not be relevant to your specific project.