You are not using a "tester". You are using a voltage sensor. Big difference.
You are getting a voltage reading but it's not working? This is almost certainly an open neutral on this circuit. A circuit needs both hot and neutral to work.
Get yourself a real tester and test from hot to neutral and hot to ground anywhere possible.
Find the open neutral and you'll find your problem.
My real advice is to call an electrician. This would be the safest bet.
You have a high resistance connection somewhere in the line. I would bet that if you have the lamp plugged into one of the outlets and turned on, if you then check another outlet, it will show a low voltage. The reason the tester and meter alone don’t show a problem is that they draw very little current.
First thing to check, with the lamp on, is voltage between neutral and ground. If it’s high, the bad connection is the neutral. If hot-neutral and neutral-ground are both low voltage, the bad connection is in the hot line.
Since you say the breaker panel was recently changed out, I’d start there. If you’re comfortable working on the panel with the cover off, check for a loose wire to the breaker or neutral bus. Remove the wires, clean or cut and restrip the wires and retighten. If you’re not comfortable, call an electrician.
One more note: you didn’t give the age of the house but if it was built in the ‘60s or ‘70s, you may have aluminum wiring. If the wires look silver colored instead of copper, stop and call an electrician. Aluminum wiring needs proper treatment to avoid future overheating and house fires. Judging from your question, I doubt you’re qualified to handle aluminum.
Best Answer
Given that your circuits are "dead", we know hot or neutral is bad. Since the magic-8-ball tester* is able to light any lights at all, we know hot must be good. Therefore neutral is bad.
Neutral broken on one single branch circuit
This is an annoying but ordinary failure. Unless you have been doing recent construction (nail through power cable), the cause is almost never the wire proper, but at the ends of the wire.
And there, the most common failure point is a cheap connection method called a "backstab", where a wire is jabbed into a hole and magic fingers grab it. They're not magic, they fail all the time just like this. This does not apply to types where you have to tighten the screw to clamp it. Most of us hunt down backstabs and kill them on sight, twisting the wires out (to preserve length) and moving the connection to a side screw or screw-to-clamp.
Neutral broken on exactly two related circuits
This is a neutral failure in a multi-wire branch circuit, which is a single circuit that carries two hots and shares 1 neutral. Like the above, it's a local problem you'll need to fix yourself, but like the below, it's a serious hazard, affecting only loads in that circuit.
Neutral broken on several circuits
Call your power company and report an outage. Now.
This is an emergency and you need to shut off all your breakers until it is fixed. Failing to do so can fry your appliances and possibly start a fire.
What's going on? North American houses are supplied ~240V power, on two poles, L1 and L2. At the midpoint (120V) a wire is brought out called "neutral". L1-neutral is 120V. L2-neutral is 120V. If you're wondering, the taller slot on your 120V sockets is neutral, the shorter slot is either L1 or L2 depending on which space the breaker is in.
Now if neutral breaks on the supply side, L1-L2 is still 240V. However Neutral is floating, which means L1-neutral and L2-neutral aren't 120V each, they only sum to 240V. Yikes: if one is significantly less than 120V, the other is significantly more and that is frying equipment and starting fires!
This is most likely on the power company, so call them up and tell them it's an outage, and tell them some guy on the Internet called it a "lost neutral". They know how urgent it is.
* So named because the "reasons" on its legend are made-up, confusing and wrong. But the three lights are useful.