In this work, we consider adaptive linear programming (ALP) decoding of linear codes over the finite field $\mathbb{F}_p$ of size $p$ where $p$ is a prime. In particular, we provide a general construction of valid inequalities for the codeword polytope of the so-called constant-weight embedding of a single parity-check (SPC) code over any prime field. The construction is based on classes of building blocks that are assembled to form the left-hand side of an inequality according to several rules. In the case of almost doubly-symmetric valid classes we prove that the resulting inequalities are all facet-defining, while we conjecture this to be true if and only if the class is valid and symmetric. For $p=3$, there is only a single valid symmetric class and we prove that the resulting inequalities together with the so-called simplex constraints give a completely and irredundant description of the codeword polytope of the embedded SPC code. For $p>5$, we show that there are additional facets beyond those from the proposed construction. We use these inequalities to develop an efficient (relaxed) ALP decoder for general (non-SPC) linear codes over prime fields. The key ingredient is an efficient separation algorithm based on the principle of dynamic programming. Furthermore, we construct a decoder for linear codes over arbitrary fields $\mathbb{F}_q$ with $q=p^m$ and $m>1$ by a factor graph representation that reduces to several instances of the case $m=1$, which results, in general, in a relaxation of the original decoding polytope. Finally, we present an efficient cut-generating algorithm to search for redundant parity-checks to further improve the performance towards maximum-likelihood decoding for short-to-medium block lengths. Numerical experiments confirm that our new decoder is very efficient compared to a static LP decoder for various field sizes, check-node degrees, and block lengths.