If in the organization of any drainage district and thereafter in the construction of ditches, drains, or other improvements, it shall be necessary to cross under or through any railroad or the right of way thereof, the same may be accomplished in the following manner: Upon filing of the report of the commissioners as to assessments and benefits and damages, they shall make a special report showing the proposed plans, manner, and character of the work as proposed in passing through such right of way, together with an estimate of the costs of same, including all damages that will be sustained by the railroad by virtue of the construction of the proposed work; and upon the hearing of such commissioner’s report, it shall be the duty of said railroad company to appear and show cause why said report should not be confirmed, as other parties interested are required to appear and on the same notice. In such showing it shall be the duty of the railroad company to file with the court, on or before the time set for the hearing, its estimate of the costs of the proposed work, including all damage that will be sustained by it by doing the proposed work. The court or chancellor in vacation shall determine the amount of such costs and damages to the railroad, and said finding shall be final and conclusive adjudication of such matters, unless appealed from in the manner provided for appeals under this chapter. The prosecution of an appeal shall not prevent the drainage commissioners constructing the work as proposed through the railroad right of way, whenever in their discretion it is necessary to do said work. Before beginning the construction of such work through such right of way, the drainage commissioners shall pay to the railroad company, or the chancery clerk for it, the amount of damages adjudicated against the district in the decree appealed from.