If you've been ambiguous in your responses to the more aggressive company to the point where they think you're ready to come on board, you may not be able to salvage their impression of you when declining. If you've said you're prepared to come in to sign the paperwork, that's quite a mixed message from what you're about to do.
Under normal circumstances, I always let people know when I am interviewing with multiple firms and I let them know I'd like to complete the process (assuming one party doesn't just drag things on and on for weeks on end) before committing to them, because I want to give them all a fair shot and come to my own conclusion about which firm is the best opportunity for me from both a financial and professional growth perspective. Some firms will react by giving you the time you ask for, and some just get more aggressive on the offer because they want to win; most of them will pressure you to answer you within X days since they want their position filled as quickly as possible.
But in your case, the best you can do is this: Say that you've unexpectedly been presented with an offer that you didn't think would come through before you needed to make a decision, and you're just waiting for the final paperwork to come through.
If you want to just focus on the preferred offer, let them know that you appreciate the offer and it's very attractive, but you think your professional interests are more closely aligned with the position at the other firm. They will be disappointed to lose out to the other firm, but most companies understand this is just how things sometimes work out.
On the other hand, if you are inclined to use the alternate offer as a negotiating lever (which may be harder if they had already been convinced they had you), you can say that you think the ACME offer is very competitive in terms of compensation and in terms of your role, and see if they open the door to further negotiation by asking you what it would take to get you to take the role.
I've been in both positions, both as an interviewer and as a candidate. It's not very surprising to me when we lose candidates to competitive offers, because the market for competent talent is fairly heated in my industry. As a candidate, I try to be as straightforward as possible without revealing my complete hand, so I wouldn't have gone the route you did. I'd decide what the minimum threshold for acceptance of either offer is, what the risks and benefits of each are, and make a gut-level decision that's based on known things, like compensation and role and apparent fit, and unknown things, like long term growth potential, apparent stability, and so on.