Both Dijkstra and A* can add different costs to the edges (=connections) from one tile to another. They also allow to connect two nodes (=tiles) with more than one edge, each one with a different cost.
The alternative jump-mode would mean that there is an alternative direct edge from each tile to each tile in jump distance. But because a mech can either walk or jump in a turn, the cost for using this edge would be the move points of a whole turn, plus the remaining points of the current turn when there already was a move this turn.
According to your description, the decision for walking vs. running does not make much of a difference regarding path choice, but it rather seem to be a strategic decisions to make. The actor definitely can walk when the destination can be reached in the current turn without resorting to running. But otherwise there are many factors to factor in, like:
- current heat level and likeliness of being involved in combat before being able to cool down
- difficulty of any shots which need to be fired this round
- how strategically important it is to reach the destination quickly
There is no hard rule for making this decision. The best you can do is to use a heuristic approach. Assign positive or negative point values to all circumstances, add them up, and see if the result is positive or negative.
There is also another factor in pathfinding you should account for: Under some conditions it could make sense for a mech to avoid ending its turn at certain locations. When in a danger area, using three turns to get from A to B but ending each one in cover might be better than using only two, but being exposed at the end of each. Or maybe not. It depends on the circumstances and the exact game mechanics. This, again, is a strategic decision you have to make based on heuristics. You could represent this by adding an additional cost to edges which end the turn on a dangerous tile to discourage the AI from making this move.