No, a humidifier will not raise the temperature enough. Some people use light bulbs put behind the cage but this system does not work very well because:
a) you need to put something dark and opaque between the cage and the light or it will hurt their eyes.
b) heat rises so you need to put the source of heat UNDER and not to the side or behind the birds.
c) you can't control the temperature with a light bulb - it will just keep on making things hotter and hotter and will end up burning whatever you put to protect their eyes.
I urge you to get a heating pad. They are not expensive and anybody who has a bird needs one, just in case BUT, if you can't afford one right now, ask relatives and friends (old people always have one for their aching bones - and I know because I am old
).
Any doxycycline will work but I doubt you will be able to get it from the pharmacy UNLESS you live in a country where it's sold over the counter (they do in my country but, in America, they force you to go to the doctor to get a prescription). They also sell doxycycline for birds online but it will take days for you to receive it and you don't have days to waste waiting for medicine for the birds (they go down VERY fast). If you happen to have any other type of antibiotic, you can use it. Plain amoxicillin or clavamox works well,too - and they are both usually prescribed for dogs and cats - but you will need to medicate in the beak which I do not recommend with little birds that are already stressed out. And, of course, there is enrofloxacin (aka Baytril), the antibiotic of choice with avian vets - but, again, it needs to be given in the beak. What you can also do is go to a petstore like Petsmart or Petco (I don't know where you are) and see what they have in terms of antibiotics for birds.
The normal dosage of doxy is 100 mg per liter (which is, more or less, a quart) but you can't make an entire quart and just use it for days, it needs to be made fresh every time so you will need to calculate the dosage for a smaller amount of water - like 1/4 of a 100 mg tablet (which you will need to crush) in a cup of water (you need to use a measuring cup). And don't worry if you can't do it exactly... only always try to make it a bit stronger rather than a bit weaker.
If you medicate exactly like I told you, the birds will get well but, if you feel that it would be better to take them to a vet, do it - only do NOT allow the vet to give it an antiobiotic injection, budgies are too small and the tissue damage done by the needle is too severe. Use only oral meds.