Put your computers on a UPS, even a small one. Just protecting yourself from short power outages will save you a bunch of downtime. Even if you resolve your overload issue, this is still worth it.
Most computer users today can get by with a laptop that's under $500, giving you built-in battery backup & portability in a compact, low-power package. Plug it in to your keyboard, mouse, and monitor the same as your desktop today, so your work experience doesn't change.
The power strip you linked to (http://www.cyberpowersystems.com/products/surge-protectors/home-surge/6050S.html?selectedTabId=specifications&imageI=#tab-box) doesn't appear to have a breaker. I think that whoever wrote that was just confused about what it means to be rated for 15A.
If your coffee maker and space heater are in the same location, you could plug them into mutually exclusive switched outlets. You'll need:
A steel square box, and an appropriate face plate
A regular duplex receptacle. 15A or 20A can work
a 3-way switch
cable
plug (15A or 20A, to match the recep)
fittings
Snap off the tab on the hot side of the recep, then run short leads from those 2 screws to the 3-way switch. This will let the switch choose one socket or the other. Plug in the heater on one and the coffee maker on the other. Now it's easy to make sure only one is in use at a time.
(Later I will add some pictures and other details. If anyone has pointers to the correct fittings and cable, please comment.)
The quickest and safest may be to check the resistance of the conductors with the power off, and determine the resistance it should be based on the material, length, and cross sectional area. Testing in various locations.
Of course this requires some math, and guessing about the conductor length.
Another quick way is to test for voltages with the power turned on. You may also need to have some sort of dummy load to check for high resistance connections, this could be done with a solenoid type voltage tester.
The easiest is to go based on the assumption that the problem is at the connections. That means that it is likely in a electrical box. Make sure that none of the connections are loose or charred in every one of the boxes. You would have still needed to do this with the other tests, but they would have helped to narrow down the location of the problem.
Actually that still isn't the easiest; which is of course to call an electrician.
I will say that it is very unlikely that you blew open the conductor anywhere else. According to a chart on Wikipedia it would take about 600 amps to melt a 14 gauge wire in one second, which is more time that it would take for a fuse to blow. So unless the cable was severely damaged beforehand somewhere along it's length; the problem is in a box.
Best Answer
But they're not individual 120V circuits!
They are combined into pairs, as something called a multi-wire branch circuit. These share a neutral, so they're not separate at all.
The handle-ties exist for a bunch of reasons.
They must be on different poles/phases
Top of the hitlist is to force them into different poles or phases, i.e. To keep someone from obliviously moving them onto a double-stuff breaker so they're on the same pole. That would set the neutral wire on fire, because the neutral wire would be handling all the current for both hots.
Whereas, if they are on different poles/phases, neutral only handles differential current: there's some math, but on a split-phase or 2-of-3-phase system, neutral will never have more current than any one hot. This is the slick trick of the multi-wire branch circuit. Take a look at a big box store's lighting sometime; they make full use of 3-phase MWBCs at 277V, and it lets them put over 20,000 watts of lighting on twelve #12 wires in a 3/4" conduit.
Anyway, as a practical matter, in modern service panels, the presence of a mandatory handle-tie forces you to position the MWBC on opposite phases/poles.
That is its most important function where someone has an itch to rearrange the panel. We'll come back to that.
Common maintenance shutoff
Typically, an electrician plugs a radio into a circuit, and shuts off breakers until the radio falls silent. This ought to work on a MWBC. Otherwise, the electrician could shut off half the MWBC, and promptly get nailed by the other half. "But how can that be? He'd only be messing with neutral, and that's harmless!" If that were true, we wouldn't insulate it... Actually neutral is hot, it's just near ground if everything is working right and properly connected.
The handle ties ensure that when he silences the radio, he has shut off the entire circuit.
This is the part of the handle-tie that is actually Code mandated. Forcing opposite poles is just a nice side effect.
But my half is tripping because of overuse of the other half!
My home is fantastically underserved. It is on half an MWBC; the other half is shared by 2 apartments. (Really.) Even so, having to reset a breaker from overload is a very rare occurrence, (maybe once every 5 years in my lifetime). It should be rare. If it's not rare, someone is doing something wrong and not taking a hint.
The cure for this is to learn about wattage of devices. Most people have no idea a hair dryer takes more power than a 54" TV. They're plainly labeled: learn what that means, learn what is on which circuits, and don't overload the circuits. My partner figured it out with only 3 explainings, and knows exactly how to sequence microwave, toaster, hair dryer and various heaters on high vs low.
But I want to overstuff my panel!
A constant refrain here is "buy a huge panel". Because it only costs a few dollars more, and it saves you from having panel-cram problems later. I gather your builder didn't get the memo.
Our quickest cure for that is "add a subpanel". I like to see a total number of full breaker spaces for a house to be in the forties. So if you have a 24 space, get another one. 16, get a 30. Etc. There are other strategies for underserved or obsolete/dangerous panels, that involve preparing the subpanel to become the new main panel when resources permit a changeover. This isn't spendthrift, this is a very economical way of being smart.
However, if you are double-stuffing this panel, here is the takeaway: they have quadplex breakers with a 2-pole breaker in the center and two individual breakers or another 2-pole on the outside. MWBCs can only go on the inside 2-pole, never the outside pair. Even if the outer ones have handle ties, which some do, those handle ties are merely decoration and do not provide common maintenance shutoff, which MWBCs require. They are labeled as such.
If you have a GE Qline panel with the cruciform bus bars which allow half width breakers, then you may use the GE half-width 2-pole breakers for an MWBC. These look like a "duplex/twin/double-stuff" breaker, but are factory handle-tied and also have common trip. You must never, never use an untied double-stuff.