What if you did not select a higher priced property than the one you sold and just wanted to pay the difference in capital gains. Could you just purchase a smaller priced property?

Yes it is allowable to buy down in value and still do an exchange.

Yes, this is referred to as trading down in value. You sell a relinquished property for one value and then acquire a replacement property of lesser value or trade down in value. It does not harm your 1031 Exchange, but you will pay taxes on the amount that you traded down by (or did not reinvest). Investors do this for strategic purposes all the time.