An even somewhat-reliable answer would require that we know the state from which this arises. Procedural rules and statutes differ dramatically from state-to-state.
My primary suggestion is that you retain new counsel if you have terminated your previous lawyer's contract. Pursuing an appeal on your own might very well result in you (1) losing; and (2) being ordered to may thousands of dollars in opposing attorney fees and/or appellate costs. I'm not a welder so I would never, ever try to manufacture my own steel grill pit. I'd probably burn my eyes, chop off a finger, and have an unusable product in the end. A non-lawyer should ever expect to pursue litigation or an appeal with legal counsel. The attorney opposing you will likely roll you quickly.
One last point. Let's say your former attorney caused you to lose $5,000 in out-of-pocket money. Given that you said it was "hundreds" of dollars, this is probably a vast over-statement of your damage model. Nevertheless, let's say it's true. It would be grossly insufficient to pursue a lawsuit against your former lawyer. As you already know, litigation is quite expensive. Further, you'd have to hire and pay a new attorney to win it. It's likely not economically advisable to sue your old attorney.
Deadlines for legal malpractice vary by state, I am presuming from your zip code, you are in the New Brunswick NJ area - and in NJ, a general rule is that you have 6 years. It is hard to tell from this fact pattern the exact amount of damages, but they do not appear to be high enough to support a claim for legal malpractice. Also, the steps you are taking now to correct the prior errors, if successful, would limit your claim even more, though on the bright side, you will have corrected it. Good luck.
Each case is fact senstive, so all answers should be viewed as general advice only, and should never replace a thorough and in depth consultation with an experienced attorney. Further, an answer should not be seen as establishing an attorney-client relationship.