You do accomplish your goal with projection, but the "making UTM uniform" or "change data to same UTM" is incorrect. A UTM zone is designed to cover a specific area with minimal distortion and that area is smaller than the US. You need a single projection designed to cover the entire US with minimal distortion - there will be more than in a UTM projection, but it cannot be avoided because you're looking at a larger area.
@AndreJ (who is a much greater expert on projections than I) has suggested a few. You do not want to use an equidistant projection - the name is a bit misleading. An equidistant projection means from the central point all distances and bearings should be to scale but not between any other two points.
There are two ways you could reproject all your different data sets. One would be to start with a new, blank map and before adding any data change the coordinate system of the data frame (Right-click in the ToC > Properties) to one as suggested by Andre. Then, each time you add a state it will be reprojected on the fly to that system (with a catch about transformation, see below). Once they are all added, you can select them all and Export to a new file. The dialog box of that process will prompt for if you want to use the data's coordinate system, the dataframe's, or one you specify. You'd pick dataframe. The other method would be the Batch Project tool, which you can add all the data sets to and perform the same project operation at once.
When you go to add a layer to your data frame that is set to one of the suggested projections you will probably get a warning message with a button called Transformations. This is probably what is causing your data to not fit together. UTM uses the WGS84 datum. The suggested coordinate systems above use NAD83. Projections that use different datums also need a transformation specified to convert between them, and it sounds like you're not specifying one. This would have to be done in the Batch Project tool as well but that doesn't give you a list to choose from.
I started a blank map, then changed the dataframe CRS to USA_Contiguous_Albers_Equal_Area_Conic. I then went to add a shapefile from those you're working with and got the following warning (left dialog):
Clicking the Transformations button leads to the dialog on the right. At the top it tells you the datum of the data being added (or already added) to the dataframe. The Into dropdown box shows the datum of the CRS the dataframe is set to. The dropdown below that is where you pick the transformation method. Click it and you get a huge list of options. The dialog tries to sort them in order that they would be most appropriate for the two inputs, so unless you know what you're doing or that you should do different, the first choice is usually a good one. I've selected
WGS_1984_(ITRF00)_To_NAD_1983. You'd have to type that in if using the Batch Project method.
Click OK, and then Close. Your state will be added to the map. Then go to add the next one. This time there will be no warning because you've already specified the transformation needed between the two datums. Your next state should come in and line up next to the previous. Once you have everything added you can convert to raster.
However I will point out this data is extremely high resolution for looking at the entire US at once. Trying to convert it to a raster at that fine of detail is going to generate a massive file and may even fail due to lack of memory. You may need to down sample the data as you create rasters (ie, use a much lower resolution grid). If you want a single raster you may need to mosaic them together. I would suggest a new question if you begin running into issues converting to raster, and leave this one as just the projection/alignment issues.