Does our technology infringe another patent?

Sadly the case is that you probably won't know until you try to file a patent at the USPTO or the PCT.

You can do searches like described here and they will give you a general sense, but I have only seen one software tool that links products to patents via semantic search and it's not available on a one-off basis (its an annual subscription).

An FTO opinion will be very costly because it is basically the law firm's reputation that is saying that there is no way you could infringe on someone else's invention at the time of the opinion. If their opinion is incorrect, then you can sue the law firm that issued the opinion (which is why they do so much work on the file and charge so much $$$).

I do patent valuations on a regular basis for a variety of companies and the price of doing a Prior Art Search (which is what you're really talking about here) generally costs way more than the cost of my actual valuation work. I suggest you do a basic search (maybe with Ashutosh's firm), file your patents if you are so inclined and then hope for the best.

Alternatively, you could do your search and not file any patents and keep your work as trade secrets... but that's for another strategic discussion.