Actually, this issue of "accord and satisfaction" is not as clear cut as one would think.
And in this case, the answer is NOT clear. Here is the problem. If you had made payment to a 'third party debt collector', it would be likely that the debt would NOT be relieved since that 3rd party would not have the authority to modify the contractual obligation that you have with the original creditor. In this case, you say that you made payment to the creditor ("The hospital cashed the check.").
The issue of 'restrictive endorsement' (as this is), hinges on the following UCC section:
"§ 3-311. ACCORD AND SATISFACTION BY USE OF INSTRUMENT.
(a) If a person against whom a claim is asserted proves that (i) that person in good faith tendered an instrument to the claimant as full satisfaction of the claim, (ii) the amount of the claim was unliquidated or subject to a bona fide dispute, and (iii) the claimant obtained payment of the instrument, the following subsections apply.
(b) Unless subsection (c) applies, the claim is discharged if the person against whom the claim is asserted proves that the instrument or an accompanying written communication contained a conspicuous statement to the effect that the instrument was tendered as full satisfaction of the claim.
(c) Subject to subsection (d), a claim is not discharged under subsection (b) if either of the following applies:
(1) The claimant, if an organization, proves that (i) within a reasonable time before the tender, the claimant sent a conspicuous statement to the person against whom the claim is asserted that communications concerning disputed debts, including an instrument tendered as full satisfaction of a debt, are to be sent to a designated person, office, or place, and (ii) the instrument or accompanying communication was not received by that designated person, office, or place.
(2) The claimant, whether or not an organization, proves that within 90 days after payment of the instrument , the claimant tendered repayment of the amount of the instrument to the person against whom the claim is asserted. This paragraph does not apply if the claimant is an organization that that sent a statement complying with paragraph (1)(i).
(d) A claim is discharged if the person against whom the claim is asserted proves that within a reasonable time before collection of the instrument was initiated, the claimant, or an agent of the claimant having direct responsibility with respect to the disputed obligation, knew that the instrument was tendered in full satisfaction of the claim."
As you can easily see, the issue of whether YOUR restrictive endorsement would meet these requirements would have to be determined by a review of the full facts (words used, placement, whether you had previously made clear of your intent to restrict, knowledge by authorized party, etc.).
Personally, I would feel that your debt was NOT relieved by your attempt to restrict.... or at least of sufficient challenge to make you spend far more than the $1800 difference in legal fees trying to support your position.
Here is a link to a site with a VERY good article on 'accord and satisfaction' and its abilities and limitations:
http://www.carreonandassociates.com/accord-satisfaction-restrictive-endorsement.htm