According to what you said,
The land does not absorb water as fast as the normal soil. Soil remains wet in the inside for a good amount of time.
It means that your soil has poor infiltration (movement of water is slow), this is a result of lack of pore space. This is common in sodic soil because sodium disperse aggregates which could results in compaction (particularly if the soil is clay-rich). To improve the structure and aggregation of soil, there is only one way, which is to apply a lot of organic amendment (and mix it well) before you plant anything (I would recommend compost). If that's a lot of work then you can do raised bed or some other variations that raises your plants above your original soil.
If you want to lower the pH (depending on which plants you wanna grow), you can add ammonium fertilizers (e.g. ammonium nitrate) which are slightly acidic. Make sure you water a lot to prevent volatilization.
If you want to lower the sodium level permanently (assuming that your water table is not high), you may be able to do so by applying calcium solution which displaces the sodium ion and the water will leach out the sodium ions. (more details can be found here: Managing Sodic Soils). But bear in mind calcium compounds won't help if you also want to lower the pH of the soil. I have a feeling potassium solution might work to displace sodium as well and it is a macronutrient for plants itself (but not entirely sure).
I think a picture or two of your soil and a detailed soil test report would help guide a better solution.