The Shipping by Weight plugin can be configured for Warehouses. In Configuration > Settings > Shipping Settings, you can select "Use warehouse location". However, it requires that you assign warehouses to products, but if you have products in "Multiple Warehouses", then nopCommerce is not smart enough to choose the warehouse closest to the customer.
The Shipping By Weight plugin can be configured for for the destination (customer) Country and State - not "city". But, you may be able to use the States to hold your cities.
But i have different shipping fees depends on locations , like from state 1 to state 2 will be 30 , from state 1 to the same state 20 How will configure this
' Shipping options could not be loaded ' usually means that the rate provider (Shipping By Weight in your case) is not returning any method/rates that match. Be sure you have a 'wildcard' record with * in all fields and a from / to range like 0 - 9999999, so that it will match if nothing else more specific matches.