You mentioned your requirements included to power a welder as well s some heavier power tools IE table saw.
Not sure how deep your pockets are, you did mention you need to get out with the least expense as possible.
Some good suggestions about hitting some auctions and engine driven welders as well.
A miller Bobcat 225 will have 8000w of power 120/240 @ 1800RPM this will have about 18 HP air cooled they could have Onan, Briggs, or even a Suburu diesel. One thing to check for if buying used is the overall condition the oil will probably have been recently changed so will look clean, ram a cloth swab as far into the exhaust as possible then examine it. Some will clean the exhaust pipe to make things look good but rarely more than a few inches inside. Black carbon soot is pretty normal but if it is oily pass it by and look at another, even heavy dry soot can mean a carburetor rebuild and tune up which shouldn't scare anyone too much if the machine will start easy and run smooth the way it is.
If buying at an auction it is difficult to put a load on the generator or the welder portion unless it has the welding leads on it but you should at least have a multimeter with you that can check AC/DC volts & HZ.
A welder/generator with only 8000 watts will have trouble pulling a 200 amp or larger 240v mig welder but will with no problems handle a CV/CC suitcase wire feeder for light duty welding.
I have several engine driven welders, a Miller Bobcat 225 a Miller big 40, a Hobart 300 gas, a Hobart champion, & a Generac 4500 to name a few not one of them will pull my Miller 210 mig, but my motor coach has a 14 KW Kubota powered by a V120BG1 diesel 21HP @1800RPM it will power the mig and both Air-conditioners, the cook top plus everything else like the TV 2 computers, the Fridge ETC. at the same time.
So a lot depends on your needs both in the short term and long term when considering your generator.