I am very sorry that you a facing such a frustrating situation. You are trying very hard to do what you feel is right, but you can't force the creditor to settle with you at a discount if they don't want to do it.
I can understand that you don't want to ruin your credit by filing bankruptcy. However, I think you already have ruined it. Not only do you have the bad credit of failing to pay on the credit cards, but they have sued you, obtained a judgement, and they are garnishing your wages and attaching your bank account.
The judgement will stay on your record for 10 years, just the same as a bankruptcy does, even if they accepted your settlement.
I think the lawyer you saw has given you sound advice.
Before you make a decision about how to proceed, you should learn more about the wage garnishment laws in your state. I am not licensed to practice law in your state, but I can tell you that federal law limits garnishment in most cases to 25%--the exceptions are for things like child support. Thus, if 25% is the maximum allowable and it's already being deducted, a second order could not increase the amount of the deduction, only the duration of the deductions. Many states also have a minimum amount you must be left with after garnishment; this is commonly 30-40 times the minimum wage. However, in some states this is calculated based on federal minimum wage, and in others it is based on the state minimum wage, which may be higher.
It's not clear whether you have additional debt that would be discharged in bankruptcy. If you have other debts that would be eliminated in bankruptcy, you may have other reasons for proceeding, but if this is the extent of your debt, you may want to negotiate further before taking that step. Figure out exactly how much of your income is subject to garnishment, and calculate how long it would take the company to recoup the full amount they're asking for. Chances are that it will be years, and it's to everyone's advantage to have this taken care of more quickly. However, since you've offered a lump-sum payment of a fairly large amount of cash, the attorneys may be assuming that you have access to more, or that you have assets they don't know about.
Sometimes it's more effective when a third party makes an offer directly, since they have no obligation on the accounts and the creditor therefore has less (really no) bargaining power.