First thing, finallyseewhy--are those savings funds SECURE? I mean he's not a joint account holder who can talk his way into getting them somehow? If not, that is job #1--get them to ANOTHER BANK and open a new account in only your name. (If it's at the joint bank, he might get lucky and get a teller who knows he's your husband and talk his way into access).
If they are secure, this is easy. He will say "I need money." You say "Are you ready to end your adultery, dedicate all of your affection and loyalty only to me, and give me access to all of your online accounts so I can verify your honesty? You're not? Oh okay call me when you are. Bye." Click. Hang up.
Don't get drawn into debates. Don't let him give you the threats and disloyal dizziness. In fact, this should pretty much be how every contact with him goes. If he calls or texts over and over, take the phone off the hook or turn off the cell. I'm sorry but you won't miss some "emergency" if you turn it off for an hour or two!
See, every time he calls, he is trying to either get a need met (like getting his ego stroked that you still want him) or pick a fight so he can blame his unhappiness on you. (See it couldn't POSSIBLY be his choices that make him unhappy--the OW is his soulmate and blah blah, so he has to blame someone. OH YEAH the evil ex-wife! That's how he justifies continuing the affair.) If he contacts you and starts in, just interrupt and say
"Are you ready to end your adultery, dedicate all of your affection and loyalty only to me, and give me access to all of your online accounts so I can verify your honesty?" Memorize it! LOL

So he may try to start ranting again or demanding or whatever, and you just say "Oh you're not? Kthxbai." Don't let him rattle you. You are not "ruining any chance your marriage might have" (he is!)--you are allowing him to learn that his choices have consequences and you are no longer PROTECTING HIM from his choices.
Come on here if he's driving you nuts. We'll be on all weekend.