We construct linear codes over odd prime fields for correcting two-dimensional Lee-metric errors on the square and hexagonal signal constellations. They are obtained by puncturing and enlarging either Reed-Solomon codes or Bose-Chaudhuri-Hocquenghem codes. We introduce two-dimensional Lee weight on the square and hexagonal constellations in a similar way as the method presented by the first author in 2019 International Symposium on Information Theory, and propose an effective method for correcting  two-dimensional Lee-metric errors of weight t>=1. The concept of value-locator of an error, which was introduced implicitly by K. Nakamura in 1979 and inherited to the above conference paper, is a key for decoding error-correcting Lee-metric codes. Our decoding method is based on the Buchberger algorithm for finding Groebner bases of ideals in the multivariate polynomial ring. The results of simulations show that our codes are more powerful for correcting two-dimensional Lee-metric errors in the two-dimensional signal constellations than the conventional codes for correcting Hamming-metric errors and one-dimensional Lee-metric errors, and the decoding method works well for correcting two-dimensional Lee-metric errors of small weight t.