yes, re-capping that radio is a good idea, but be aware that you will most likely need to re-align the radio afterwards, and that takes good test equipment.
what you are describing sounds like you have lost a voltage regulator.
so, let's start with what does work.
the radio lights up, which im guessing means that the meter light comes on, and the channel display comes on.
if the channel display does not come on, please correct me.
you said that you can hear the relay click when keying the mic, but that you have no TX power on AM and no receive.
please do check if you have any SSB audio by talking into the mic on SSB TX and looking at a wattmeter or the radio's meter.
we will start tracing power, so you'll need a voltmeter set to read DC volts in a range that goes up to 14 volts or so. if you have a probe with an alligator clip on it, use it for the negative lead and clip that lead to PC board ground since you will be making multiple checks and it will be easier to just use the positive to probe around the board.
any metal tuning can will make a good connection point, as does the negative pin coming in to the back of the radio.
just don't use the chassis, as it is isolated from DC ground.
we will use this schematic to trace power:
http://www.cbtricks.com/radios/cobra/138xlr/graphics/cobra138xlr_sch.pdf
lets start at the lower right corner where the power comes in to the radio.
starting at the positive pin, you will measure between 12-14 volts here or whatever your power supply is set at. it should be 13.8 volts.
what we will be looking for, is either a transistor or an IC chip that has this 13.8 volts (also known as VCC or supply voltage) on one leg, but does not have the appropriate lower voltage on another leg.
your PLL chip runs on 5 volts, so there will be one transistor that has VCC on one leg, ground in the middle, and 5 volts on the other leg as an example.
if this 5 volts were missing, your PLL would not work and you would have no receive or transmit.
ok, follow that positive line through L40, through S406 (the power switch), and notice that the relay coil is connected to this line.
the other side of that relay coil needs to go to PC board ground in order to engergize the relay and put the radio in to TX mode.
if the CB/PA switch is in CB position, and you follow it up the schematic, you will notice that it goes to pin 3 of the mic jack. that pin gets grounded when you key the mic, so, if your relay clicks, you can say that you have good power to the relay, and the CB/PA switch is working. well, at least in CB mode which is what we care about.
ok back to that positive voltage line we are following.
we know the problem exists in both RX and TX so it's not going to be that RX TX switch that you see.
so let's follow that voltage line on the schematic up past R113 to where it moves to the left, all the way across the schematic, then up again connecting to R161 which is just to the right of the UPD858 PLL chip.
measure the voltage on the other side of R161 which should be just a bit lower than the incoming voltage.
then follow that line up to IC6 which is the 5 volt regulator to the PLL chip.
if the transistor is facing you and upright, pin 1 is the input pin and is on the left.
the middle pin is ground, and the pin on the right is the output 5 volts.
so measure the voltage on the input pin, and then check for the 5 volts on the output pin.
is it there?
we will stop there for now.
please post back the results of your tests.
LC