In a perfect world, you would sit down with the chair and map out a contingency plan like two reasonable adults. Barring that, I'd suggest you wait to tell your chair until the outside offer is definite, and then (assuming you are willing and able) offer to swap your scheduled section for an evening one so as not to leave them in the lurch. That way you give the chair the choice between 1) losing a section, 2) asking/forcing a full-time faculty member to swap times, and 3) having two evening sessions, hopefully not on the same night of the week. Either way s/he goes, you have been reasonable and accommodating. And by all means, tell your chair right away that you are looking for full-time jobs (as any adjunct should be), and ask him/her to write a letter of recommendation for your file.
Good luck!
In an improved world, offers from colleges and the government would be serious, solid offers and the recipient would be able to give ample notice to whichever offer he has entertained but now declines so that they can make a carefully considered choice for a replacement.