It seems to me that if your group recently was dealing with an embezzling situation, that you (collectively as a board) need to be extra careful about how the $$ is handled. You (all - especially the Pres) should be willing to go the extra mile to ensure the money trails are visible, auditable, authorized and accounted for. A group in this situation should be very careful about authorizing expenditures.
I understand your annoyance - people who make rules for the group but don't follow them is a pet peeve of mine. They set a bad example and tell the rest of the group not to take their responsibilities seriously. Well, you and the other new board member need to be serious.
I understand there is friction between you two, but you need to find away to work around that. If you talk with the Pres and remind her of the embezzeling situation and the new rules that are in effect, hopefully that works. If it doesn't you probably need to talk to the other board members and the Principal and work together to turn the ship in the right direction. If you stick to facts and don't bring pettiness or personalities into it, you will probably be successful (eventually).
Our group has a budget and the Treasurer is responsible for tracking to that budget. The Treasurer is NOT authorized to write a check that will exceed any budget item but is authorized to reimburse for appropriate expenditures within the budget item without specific board approval.
For example, say our Social Activities budget is $700. That budget item typically pays for things like plastic-ware and paper plates that will be shared by multiple events. If the committee chair finds a great deal on the stuff and spends, say $200, the Treasuer can reimburse that without a vote - it is an expenditure that is part of the rules. (This is not to say that the committee chair and Treasurer don't let the rest of the board know what happened - they obviously should and they have.) if, say, there were only $100 left in the budget, then the Committee Chair would ask for an increase in budget at the next meeting or the Treasurer would ask for the increase from the Board (depending on timing). Worst case, the committee chair would have to return half the stuff.
With e-mail we can get a consensus and vote on emergency items pretty quickly. There are very few emergencies that can't wait a day or so to be paid for.
It is not a perfect system (we are working on documenting the rules for each budget item), but it works for us.
I am very wary of a blank check given to any PTO member, dlf, no matter how trustworthy. It sets a bad precedent and allows a less trustworthy successor to take advantage.
Having said all that, writermom, sometimes it is just hard to get to the bank for a few days, so depending on the circumstances I can cut your Pres a little slack that the money wasn't deposited right away. I would expect that it would be kept in a safe, secure, non-visible place in the home rather than being left in the car while running errands though.
Good luck