Your current amp will give 2x 350W to 8Ohm and while this sound ok, there is a chance of damage. Usually it is best to have 20-30% headroom on your amp. Amps that are run at full signal start to get worse as they near their limits. At this point it is possible for the signal to become clipped to form a block signal rather than a smooth sinus signal. And especially high end drivers (tweeters for you hifi folks out there) can get damaged if they get a block signal. So in your case I’d keep my amp somewhere around 85-90% of max to prevent that from happening.
You should run sub-woofers on a separate amp. And it should indeed be able to deliver at the very least 2x 600W into 4 Ohm, preferably in the 700/750W range though.
That said, you now also need to have an active cross-over. And unfortunately there aren’t many of those that are affordable and let you just split at the subwoofer level. So you either can’t or you’ll have to do a lot of tweaking.
This (and the whole calculating ohms and all that) is the reason we are very strong advocates of using active speakers and not amp/passive speaker combinations. In the old(er) days, amp/passive was the better-sounding option. But today with all the modern amp technology, if you buy brand gear, you will get sound that is at least as good as the amp/speaker alternative and it’s much simpler and easier to hook up. Also since you now have two amps for two speakers, if one unit dies, you have a second one to keep you going, so built-in redundancy. Adding more speakers (bigger event) is as easy as running an extra set of XLR cables from your first speakers throughput/output to the next set and off you go.
I can understand if you don’t want to or can’t get rid of the amp/speakers and move to active for the full-range. What you CAN do is get decent ACTIVE sub-woofers. They will normally have high-pass filters built-in (x-over).
You would then send the signal from your controller master output (by XLR) to a sub-woofer. Then take the sub-woofer high-pass output (so not the full range one, you don’t want your full range speakers to also play the frequencies that the sub is now handling) and run another XLR cable to the input of your amp. Then you can start saving up to replace the amp/speakers in the future.
Hope that helps some.