If I wrote” Paid in Full” on a $25 check for a debt of $450, if the check is cashed does that constitute acceptance of full payment?

It does not--otherwise, we'd all do it. For there to be an agreement or contract--such as an agreement to settle a debt for less than it's full amount--there consideration. In the case you describe there is no "consideration," which is getting something for giving something; since you already owe $450, the creditor has a right to receive $450 of payments. That means that paying him the $25 doesn't give him anything beyond what's already "his"--i.e. anything beyond what he's already entitled to. Therefore, there is no consideration to bind the agreement and the creditor can cash the check and go after you for  the rest.

That's not to say that a creditor cannot voluntarily agree to settle a debt for less than full payment. But that must be done by a mutual agreement, showing that both parties have negotiated to the outcome and there is a meeting of the minds. One party cannot unilaterally impose an agreement on another, especially--as indicated above--where the creditor has every right to simply apply the check vs. the balance outstanding. If you want to settle a debt for less than full, you need to talk to the creditor, negotiate with it, and get mutually agreement clearly indicated in writing.

Try first getting the other party to agree to accepting less than full payment; then send them the payment once you have their agreement.

It's a good thing this is the rule, btw--otherwise, if you sold a car or home, the buyer might do this to you; or your boss might try to settle your last paycheck whenever you leave for  just dimes on the dollar; etc.