Hypothetically, would digging a big trench from the ocean and flooding desert areas be a bad solution to combatting sea level rise?
Yes, it would be a bad solution.

Aside from the construction and maintenance cost, the impact to the populations affected, and the impact on the environment around the destination, the benefits would be minimal, as roughly calculated in the answer to Volume of Land Below Sea Level


  So to reduce the sea level by 1m the total area of these depressions
  needs to be around 5,000,000 km^2. This is around half the total land
  area of the USA. I don't have the figures to hand but given that these
  area would probably all fit into an area considerably smaller than
  Texas - I estimate that flooding land which is below sea level will
  reduce the sea level by exactly 'not very much at all' (give or take a
  millimeter - and very much less than 1m in any event).


There is also a rate of flow issue. You basically want to create rivers flowing inland at a rate to offset or reverse sea level rise, which means the flow rate of those rivers has to approach the volume of global sea rise. Global sea level is rising by an average of about 3.4mm per year, or 1,230 cubic kilometers per year according to NASA (and the rate is increasing). That amount is about the flow of the Congo River (the second largest river in the world) or roughly 19 percent of the flow of the Amazon River.

Fed by gravity, these are among the widest and deepest rivers of the world; if you're going to move that volume of water, you need to build channels to each basin with a net flow similar to that capacity, with that gradient, that won't leach saltwater into the groundwater of surrounding communities. (The engineering issues likely pale in comparison to the political challenges.)

And keep in mind that transferring water inland is only a temporary solution. Water evaporates and by spreading water over a larger area you're increasing its surface area; Death Valley has an evaporation rate of 140 inches or 3.5 meters per year. (In Southern California, the Salton Sea is drying out and causing a variety of salinity and dust issues due to lack of inflows.) So you're transferring ~60-degree seawater into a salt pan where it's going to heat up and evaporate and leave behind a toxic brine, then fall or flow back into the ocean, contributing in some small way to the thermal expansion of the ocean.

(Also, the net volume of CO2 released by building and maintaining these massive projects would also contribute to global warming in some non-negligible way.)